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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

Give an example of a practical situation where the calculation nPr = 120 might arise.
Ten cars are to be parked in a car park that has 20 parking spaces set out in two rows of 10. Find how many different patterns of unoccupied parking spaces are possible if:a. The cars can be parked
a. How many distinct arrangements of the letters in the word STATISTICS are there?b. Find the probability that a randomly selected arrangement begins with:i. Three Ts ii. Three identical letters.
Twenty teams have entered a tournament. In order to reduce the number of teams to eight, they are put into groups of five and the teams in each group play each other twice. The top two teams in each
a. Under what condition is nPr > nPn-r?b. Given that nPr × nPn-r = k × nPn, find an expression for k in terms of n and r.
A boy has eight pairs of trousers, seven shirts and six jackets. In how many ways can he dress in trousers, shirt and jacket if he refuses to wear a particular pair of red trousers with a particular
Three skirts, four blouses and two jackets are hung in random order on a clothes rail. Find the probability that:a. The three skirts occupy the middle section of the arrangementb. The two jackets are
A bank provides each account holder with a nine-digit card number that is arranged in three blocks, as shown in the example opposite. Find, in index form, the number of card numbers available if:a.
Five playing cards are randomly selected from a standard deck of 52 cards. These five cards are shuffled, and then the top three cards are placed in a row on a table. How many different arrangements
A girl has 11 objects to arrange on a shelf but there is room for only seven of them.In how many ways can she arrange seven of the objects in a row along the shelf, if her clock must be included?
In a group of 180 people, there are 88 males, nine of whom are left-handed, and there are 85 females who are not left-handed. If six people are selected randomly from the group, find the probability
A basket holds nine flowers: two are pink, three are yellow and four are red. Four of these flowers are chosen at random. Find the probability that at least two of them are red.
Seven chairs, A to G, are arranged as shown.In how many ways can the chairs be occupied by 7 of a group of 12 people if three particular people are asked to sit on chairs B,D and F, in any order?
A Mathematics teacher has 10 different posters to pin up in their classroom but there is enough space for only five of them. They have three posters on algebra, two on calculus and five on
A small library holds 1240 books: 312 of the 478 novels (N) have hard covers (H), and there are 440 books that do not have hard covers. Some of this information is shown in the Venn diagram
Find the number of ways in which 11 different pieces of fruit can be shared between three boys so that each boy receives an odd number of pieces of fruit.
A minibus has 11 passenger seats. There are six seats in a row on the sunny side and five seats in a row on the shady side, as shown in the following diagram. Find how many ways eight passengers can
As discussed at the beginning of this chapter in Explore 5.1 about encrypting letters, it states that there are over 27 million possibilities for the password encrypted as UJSNOL. How many
A netball team of seven players is to be selected at random from five men and 10 women. Given that at least five women are selected for the team, find the probability that exactly two men are
A bakery wishes to display seven of its 14 types of cake in a row in its shop window. There are six types of sponge cake, five types of cheesecake and three types of fruitcake. Find the number of
How many distinct three-digit numbers can be made from 1, 2, 2, 3, 4 and 5, using each at most once?
Two items are selected at random from a box that contains some tags and some labels. Selecting two tags is five times as likely as selecting two labels. Selecting one tag and one label is six times
Five cards, each marked with a different single-digit number from 3 to 7, are randomly placed in a row. Find the probability that the first card in the row is odd and that the three cards in the
From three sets of twins and four unrelated girls, find how many selections of five people can be made if exactly:a. Two sets of twins must be included b. One set of twins must be included.
A photograph is to be taken of a pasta dish and n pizzas.The items are arranged in a line in random order.Event X is ‘the pasta dish is between two pizzas’.a. Investigate the value of P(X ) for
Two ordinary fair dice are rolled and the two faces on which they come to rest are hidden by holding the dice together, as shown, and lifted off the table.The sum of the numbers on the 10 visible
Three ordinary fair dice are rolled. Find the number of ways in which the number rolled with the first die can exceed the sum of the numbers rolled with the second and third dice. Hence, find the
How many even four-digit numbers can be made from the digits 0, 2, 3, 4, 5 and 7, each used at most once, when the first digit cannot be zero?
a. i. Find how many numbers there are between 100 and 999 in which all three digits are different.ii. Find how many of the numbers in part i are odd numbers greater than 700.b. A bunch of flowers
Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements ifi. The first and last cans in the row
Each of the eight players in a chess team plays 12 games against opponents from other teams. The total number of wins, draws and losses for the whole team are denoted by X, Y and Z, respectively.a.
Six books are randomly given to two girls so that each receives at least one book.a. In how many ways can this be done?b. Are both girls more likely to receive an odd number or an even number of
The 60 members of a ballroom dance society wish to participate in a competition but the coach that has been hired has seats for only 57 people. In how many ways can 57 members be selected if the
Four discs in two colours and in four sizes are placed in any order on either of two sticks. The following illustration shows one possible arrangement of the four discs.a. Find the number of ways in
Without using a calculator, find the value of:a. 5!/3!b. 4!/2! – 3!c. 7 × 4! + 21 × 3!d. 10!/8! + 9!/7!e. 20!/18! – 13!/11!
In how many ways can the six letters A, B, C, D, E and F be arranged in a row?
Find the number of distinct arrangements of all the letters in these words:a. TABLE b. TABLET c. COMMITTEEd. MISSISSIPPI e. HULLABALLOO.
Find how many five-digit numbers can be made using the digits 2, 3, 4, 5 and 6 once each if:a. There are no restrictionsb. The five-digit number must be:i. Odd ii. Even iii. Odd and less
Find how many permutations there are of:a. Five from seven distinct objects b. Four from nine distinct objects.
Find the number of ways in which five apples can be selected from:a. Eight apples b. Nine apples and 12 oranges.
Two children are selected at random from a group of six boys and four girls. Use combinations to find the probability of selecting:a. Two boysb. Two girlsc. One boy and one girl.
The word MARMALADE contains four vowels and five consonants. Find the number of possible arrangements of its nine letters if:a. There are no restrictions on the orderb. The arrangement must begin
Use your calculator to find the smallest value of n for which:a. n! > 1000000b. 5! × 6! < n!c. (n!)! > 1020
From a standard deck of 52 playing cards, find how many ways there are of arranging in a row:a. All 52 cards b. The four kings c. The 13 diamonds.
Find how many six-digit numbers can be made from these sets of digits:a. 1, 1, 1, 1,1 and 3 b. 2, 2, 2, 7, 7 and 7 c. 5, 6, 6, 6, 7 and 7 d. 8, 8, 9, 9, 9 and 9.
Find how many ways four men and two women can stand in a line if:a. The two women must be at the frontb. There must be a woman at the front and a man at the backc. The two women must be separatedd.
From 12 books, how many ways are there to select and arrange exactly half of them in a row on a shelf?
From seven men and eight women, find how many ways there are to select:a. Four men and five women b. Three men and six women c. At least 13 people.
Three chocolates are selected at random from a box containing 10 milk chocolates and 15 dark chocolates. Find the probability of selecting exactly:a. Two dark chocolates b. Two milk
Five men, four children and two women are asked to stand in a queue at the post office. Find how many ways they can do this if:a. The women must be separatedb. All of the children must be separated
Use your calculator to find the largest value of n for which:a. n!/500000 < 80b. 1.5 × 1012 – n! > 0c. n!/(n – 2)! < 500
In how many different ways can the following stand in a line?a. Two women b. Six men c. Eight adults.
A girl has 20 plastic squares. There are five identical red squares, seven identical blue squares and eight identical green squares. By placing them in a row, joined edge-to-edge, find how many
Find the ratio of odd-to-even six-digit numbers that can be made using the digits 1, 2, 3, 4, 5 and 7.
In how many ways can gold, silver and bronze medals be awarded for first, second and third places in a race between 20 athletes? You may assume that no two athletes tie in these positions.
a. How many different hands of five cards can be dealt from a standard deck of 52 playing cards?b. How many of the hands in part a consist of three of the 26 red cards and two of the 26 black cards?
Four bananas are randomly selected from a crate of 17 yellow and 23 green bananas. Find the probability that:a. No green bananas are selected b. Less than half of those selected are green.
Find the probability that a randomly selected arrangement of all the letters in the word PALLETTE begins and ends with the same letter.
Express, in as many different ways as possible, the numbers 144, 252 and 1 1 2 in the form a! × b!/c!, where none of a, b or c is equal to 0 or to 1.
In how many different ways can the following sit in a row on a bench?a. Four girls b. Three boys c. Four girls and three boys.
Two students are asked to find how many ways there are to plant two trees and three bushes in a row. The first student gives 5! = 120, and the second gives 5!/2! × 3! = 10. Decide who you agree with
Find how many ways 10 books can be arranged in a row on a shelf if:a. The two oldest books must be in the middle two positionsb. The three newest books must not be separated.
a. Find the number of ways in which Alvaro can paint his back door and his front door in a different colour if he has 14 colours of paint to choose from.b. In how many ways could Alvaro do this if he
a. From the 26 letters of the English alphabet, find how many ways there are to choose:i. Six different lettersii. 20 different letters.b. Use your results from part a to find the condition under
A curator has 36 paintings and 44 sculptures from which they will randomly select eight items to display in their gallery. Find the probability that the display consists of at least three more
Eight-digit mobile phone numbers issued by the Lemon Network all begin with 79.a. How many different phone numbers can the network issue?b. Find the probability that a randomly selected number issued
Express the area of a 53cm by 52cm rectangle using factorials.
Seven cars and x vans can be parked in a line in 39 916 800 ways. Find the number of ways in which five cars and x + 2 vans can be parked in a line.
Ten coins are placed in a row on a table, each showing a head or a tail.a. How many different arrangements of heads and/or tails are possible?b. Of the arrangements in part a, find how many have:i.
Five cows and one set of twin calves can be housed separately in a row of seven stalls in 7P7 =5040 ways. Find in how many of these arrangements:a. The two calves are not in adjacent stallsb. The two
In a classroom there are four lights, each operated by a switch that has an on and an off position. How many possible lighting arrangements are there in the classroom?
There are 12 books on a shelf. Five books are 15cm tall; four are 20cm tall and three are 25cm tall. Find the number of ways that the books can be arranged on the shelf so that none of them is
Two cubical boxes measure 25cm by 24cm by 23cm, and 8cm by 7cm by 6cm. Express the difference between their volumes using factorials.
A woman has 10 children. She arranges 11 chairs in a row and sits on the chair in the middle. If her youngest child sits on the adjacent chair to her left, in how many ways can the remaining children
There are 420 possible arrangements of all the letters in a particular seven-letter word. Give a description of the letters in this word.
Find how many of the six-digit numbers that can be made from 1, 2, 2, 3, 3 and 3:a. Begin with a 2 b. Are not divisible by 2.
From a group of 10 boys and seven girls, two are to be chosen to act as the hero and the villain in the school play. Find in how many ways this can be done if these two roles are to be played by:a.
From six boys and seven girls, find how many ways there are to select a group of three children that consists of more girls than boys.
In a toolbox there are 25 screwdrivers, 16 drill bits, 38 spanners and 11 chisels. Find the probability that a random selection of four tools contains no chisels.
The 11 letters of the word REMEMBRANCE are arranged in a line.i. Find the number of different arrangements if there are no restrictions.ii. Find the number of different arrangements which start and
Eight children each have seven boxes of six eggs and each egg is worth $0.09. Write the total value of all these eggs in dollars, using factorials.
A group of n boys can be arranged in a line in a certain number of ways. By adding two more boys to the group, the number of possible arrangements increases by a factor of 420. Find the value of n.
An author has written 15 children’s books. The first eight books that she wrote contained between 240 and 250 pages each. The next six books contained between 180 and 190 pages each. Correct to 1
a. Find the mean and standard deviation of the first seven positive even integers.b. Without using a calculator, write down the mean and standard deviation of the first seven positive odd integers.c.
Over a short period of time in 2016, the value of the pound sterling (£) fell by 15.25% against the euro (€). Find the percentage change in the value of the euro against the pound over this same
120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.Calculate estimates of the
A company manufactures right-angled brackets for use in the construction industry. A sample of brackets are measured, and the number of degrees by which their angles deviate from a right angle are
The mass of waste produced by a school during its three 13-week terms is given in tonnes, correct to 2 decimal places, in the following table.a. Calculate estimates of the mean and standard deviation
A set of n pieces of data has mean x and standard deviation S.Another set of 2n pieces of data has mean x and standard deviation ½ S.Find the standard deviation of all these pieces of data together
Twenty values of x are summarised by Σ(x −1)2 = 132 and Σ(x −1) = 44.Eighty values of y are summarised by Σ(y +1)2 = 17 704 and Σ(y +1) = 1184.a. Show that Σx = 64 and that Σx2
The weights, in kilograms, of the 15 basketball players in each of two squads, A and B, are shown below.i. Represent the data by drawing a back-to-back stem-and-leaf diagram with squad A on the left
The following table shows the cumulative frequencies for values of x.Without drawing a cumulative frequency graph, find:a. The interquartile rangeb. The 85th percentile.
Twenty values of x are summarised by Σ(x −1)2 = 132 and Σ(x −1) = 44.Eighty values of y are summarised by Σ(y +1)2 = 17 704 and Σ(y +1) = 1184.a. Show that Σx = 64 and that Σx2
The mass of waste produced by a school during its three 13-week terms is given in tonnes, correct to 2 decimal places, in the following table.a. Calculate estimates of the mean and standard deviation
The heights, xcm, of a group of 82 children are summarised as follows.Σ(x −130) = −287, standard deviation of x = 6.9.i. Find the mean height.ii. Find Σ(x −130)2.
Fifty 10-gram samples of a particular type of mushroom are collected by volunteers at a university and tested. The following table shows the mass of toxins, in hundredths of a gram, in these
Refer to the following diagram. In position 1, a 10-metre rod is placed 10 metres from a fixed point, P. Six small discs, A to F, are evenly spaced along the length of the rod. The rod is rotated
The heights, x cm, of 200 boys and the heights, y cm, of 300 girls are summarized by the following totals:Σ(x −160)2 = 18 240, Σ(x −160) = 1820, Σ(y −150)2 = 20 100, Σ(y −150) =
A sample of 36 data values, x, gave Σ(x − 45) = −148 and Σ(x − 45)2 = 3089.i. Find the mean and standard deviation of the 36 values.ii. One extra data value of 29 was added to the
A 9-year study was carried out on the pollutants released when biomass fuels are used for cooking. Researchers offered nearly 1000 people living in 12 villages in southern China access to clean

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