1. Consider the game Yahtzee and a single roll of 5 dice. For this problem, assu
ID: 3329717 • Letter: 1
Question
1. Consider the game Yahtzee and a single roll of 5 dice. For this problem, assume the order
of the dice does not matter. Thus, ultimately, each roll corresponds to choosing r = 5 items (dice)
from as set of n = 6 values. Thus, the total number of different rolls possible is given by the formula
n+r1Cr = 10C5 = 252. For the following questions, re-apply this formula where applicable and simplify
your answers to decimal form.
(a) How many different rolls have no repeated value (they are all unique)?
(b) How many different rolls correspond to a full house?
(c) How many different rolls correspond to a large straight?
(d) How many different rolls correspond to a small straight but not a large straight?
(e) How many different rolls have exactly three 5’s?
(f) How many different rolls contain a three-of-a-kind, but not a 4-of-a-kind?
2.Prove that in a classroom of n > 1 students, there are at least 2 students who know the same number of other students. You can assume that if student A knows student B, then student B knows student A.
3. How many passwords of exactly 7 lower case letters contain: (a) the letter q? (b) the letters x and z? (c) the letters x and y with x appearing before y, with all letters distinct?
Explanation / Answer
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3) a) Assuming are letters are distinct:
Number of passwords = 25C6 * 7!
b) Letters 'x' and 'z'
Number of passwords = 24C5 * 7!
c) 'x and 'y'
Number of passwords = 24C5 * ways that 'x' appears before 'y'
Ways that 'x' appears before 'y' = 6! + 5C1 * 5! + 4C1 * 5! + 3C1 * 5! + 5!
So total = 24C5 * (! + 5C1 * 5! + 4C1 * 5! + 3C1 * 5! + 5!)
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