Discrete Mathematics: The Principles of Counting/Combinatorics Let d be a positi

Question

Discrete Mathematics: The Principles of Counting/Combinatorics

Let d be a positive integer. Show that among any group of d + 1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d. A professor teaching a Discrete Math course gives a multiple-choice quiz that has ten questions, each with four possible responses: a, b, c, d. 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 must be identical? (Assume that no answers are left blank.) How many different ways are there to choose a dozen donuts from the 21 varieties at a donut shop?

Explanation / Answer

Solution:-

Question(20):-->

Let d be positive integer .

let n be anothe integer.

Let ni , for i = 1, . . . , d + 1 be the d + 1 integers.

Let ri , for i = 1, . . . , d + 1 be their respective remainders, when divided by d.

Since there are d+ 1 remainders whose values must be in the interval [0, d1], there must be 2 remainders of the same value.

Their corresponding integers, thus have the same reminder when divided by d.

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