(i) State the general form for the expansion of (w+x+y+ z) according to the multinomial theorem....

  

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(i) State the general form for the expansion of (w+x+y+ z)" according to the multinomial theorem. (ii) What is the sum of all numbers of the form 11! a!b!c!d!' " where a, b, c and d range over all nonnegative integers satisfying a+b+c+d=11. (b) A professor teaching a university course gives a multiple choice quiz that has five questions, each with four possible responses: a, b, c, d. Assuming that no answers are left blank, what is the minimum number of students that must be in the professor's class in order to guarantee that at least three answer sheets are identical? (c) On a table, there is a basket of fruits containing an apple, an orange, a pear, 2 kiwis and 3 mangosteens. Suppose you want to choose five fruits from the basket. However, the kiwis are in a packet where they must be taken as a pair and the mangosteens are altogether in a string net bag where they must be taken as a triple. Use generating function to determine how many ways the choice can be made. (i) State the general form for the expansion of (w+x+y+ z)" according to the multinomial theorem. (ii) What is the sum of all numbers of the form 11! a!b!c!d!' " where a, b, c and d range over all nonnegative integers satisfying a+b+c+d=11. (b) A professor teaching a university course gives a multiple choice quiz that has five questions, each with four possible responses: a, b, c, d. Assuming that no answers are left blank, what is the minimum number of students that must be in the professor's class in order to guarantee that at least three answer sheets are identical? (c) On a table, there is a basket of fruits containing an apple, an orange, a pear, 2 kiwis and 3 mangosteens. Suppose you want to choose five fruits from the basket. However, the kiwis are in a packet where they must be taken as a pair and the mangosteens are altogether in a string net bag where they must be taken as a triple. Use generating function to determine how many ways the choice can be made.