Question

Suppose that a “word” is any string of six letters. Repeated letters are allowed. For our purposes, vowels are the letters a, e, i, o, and u. a) How many words are there? b) How many words begin with a vowel? c) How many words begin with a vowel and end with a vowel? d) How many words have no vowels? e) How many words have exactly one vowel?

A professor teaching a Discrete Math course gives a multiple choice quiz that has six
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.)

Answer 1

Question 1:

a) Number of words = 26^6 [since each letter can be selected from one of the 26 characters a-z]

b) Number of words being with vowel = 5 * 26^5 [ since the first place must be filled with a vowel, which are 5 (a,e,i,o,u), rest 5 places can be filled by any alphabet]

c) Number of words being with vowel and end with vowel = 5^2 * 26^4 [since the first and last place must be filled with vowel]

d) Number of words with no vowel = 21^6 [we are left with only (26-5) = 21 possibilities for each spot after removing the 5 vowels from the queue]

e) Number of words having exactly one vowel = There are 6 possible places to place the vowel, which can be done in 6C1 ways, then the vowel can be selected in 5 ways and remaining 5 positions can be filled in 21^5 ways

Total ways = 6C1 * 5 * 21^5

Question 2:

Since the test contains 6 questions, hence we can answer the first problem in 4-ways, second problem in 4-ways and so on for 3rd, 4th, 5th and 6th ways

Number of ways to answer the quiz = 4^6 = 4096 ways

This is the problem of pigeonhole principle, now we have 4096 holes, which are the particular set of answers to the quiz.

Now for getting same three answer sheets, we need to fill all the holes first twice, which is 4096 * 2 = 8192

Now, the next student must belong to any of the hole, so one hole must contain 3 similar answer sheets.

Hence the minimum number of students must be in professor class will be equal to 8193

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