Q1. The game of Motzhee consists of rolling five 8-sided dice (each die of a dif
ID: 3204790 • Letter: Q
Question
Q1. The game of Motzhee consists of rolling five 8-sided dice (each die of a differing color), each of whose faces are numbered with the integers {1, 2, 3, 4, 5, 6, 7, 8}. What is the probability of rolling a full house? A full house consists of five dice, three of which have the same number facing up, the other two of which have the same number facing up as well, but where the number facing up in the triple is different from that of the pair.
Q2. Show that (p ( p (( ~q ) s))) (p ((~q) s)) p.
Q3. In how many different ways can one choose n out of 2n objects, given that n of the 2n objects are identical and the other n are all unique?
Q4. The working days in the next year can be numbered 1,2,…,300. Homer wants to avoid as many as possible.
• On even-numbered days, Homer will say he’s sick.
• On remaining days that are a multiple of 3, he will say he’s stuck in traffic.
• On remaining days that are a multiple of 5, he will refuse to come out from under the blankets.
In total, how many work days will he avoid in the coming year?
Q5. How many of the billion numbers in the range from 1 up to and including 1,000,000,000 contain the digit 1?
Q6. Consider the set of n-digit sequences of digits 0,1,…,9. Two sequences are said to be of the same type if the digits of one are a permutation of the digits of the other. How many types of the 10^n, n-digit sequences are there?
Q7. How many ways are there to order the 26 letters of the alphabet so that no two of the vowels a, e, i, o, u appear consecutively and the last letter in the ordering is not a vowel?
Explanation / Answer
(1)
If 5 8-sided dice rolled, total possible outcomes are 8^5.
In order to have full house, first 3 dice should have the same number hence probability of having the same number on first 3 dice = (1/8)^3
remaining 2 dice should show the same number but different from the first 3, hence probability of this = (1/7)^2
Hence required probability = (1/8)^3*(1/7)^2
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