Problem1 In this question, there are 5 identical apples and 6 identical oranges to be distributed. (a) How many ways are there to distribute the fruit among 4 children? It is allowed for some child or children to receive nothing.(b) How many ways are there to distribute the fruit among 4 children if each child needs to receive at least one orange and not all of the fruit is required to be handed out? (c) How many ways are there to distribute the fruit among 3 children such that no child receives two different types of fruit?It is allowed for some child or children to receive nothing (d) How many ways are there to lay the fruit in 3 piles? The piles must not be empty but it doesn't matter which pile is which. (e) If one of the 3-pile arrangements from part (d) is chosen uniformly at random, what is the probability that each pile consists of only one type of fruit?

Problem 2A random permutation of n objects is selected.

3 pts(a)What is the probability that no pair of objects lies in their initial positions

(either exactly as they were or swapped) if n = 3?

3 pts(b)What is the probability that no pair of objects lies in their initial positions

(either exactly as they were or swapped) if n = 4?

4 pts(c)Let n be an even number, say n=2k. Let L be the (random) number of

odd-numbered objects that are placed in odd-numbered positions.Give the name of the distribution

of L, including

ters. Also state the mean and variance of L.