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### PP-exam1

Course: STOR 215, Fall 2007
School: UNC
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215, STOR Fall 2007 Practice Problems for Midterm I: Questions in Old Exams 1. (a) How many arrangements are there of the letters in the word SHENANIGAN? (the 'A' letters are considered indistinguishable from each other, and the 'N' are indistinguishable from each other) (5 points) (b) How many of these are such that there are no 'N' letters next to each other? (5 points) 2. Let k and n be integers, 1 k n....

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215, STOR Fall 2007 Practice Problems for Midterm I: Questions in Old Exams 1. (a) How many arrangements are there of the letters in the word SHENANIGAN? (the 'A' letters are considered indistinguishable from each other, and the 'N' are indistinguishable from each other) (5 points) (b) How many of these are such that there are no 'N' letters next to each other? (5 points) 2. Let k and n be integers, 1 k n. Prove combinatorially n n-1 n-1 = + . k k k-1 (5 points) 3. Draw Pascal's triangle down to row 5. Mark the location of in it. (5 points) 4. We know that (n + 1)(n + 2)(2n + 3) n(n + 1)(2n + 1) + (n + 1)2 = 6 6 (so you do NOT need to prove this, you can use it without proof). Prove that for all positive integers i=1 5 n, n 4 and 1 3 i2 = n(n + 1)(2n + 1) 6 (7 points) 5. Given a regular deck of 52 cards, how many 13 card hands are there, which have 3 pairs AND 3 other cards of the same rank (but different from the rank of the pairs), AND 4 remaining cards of the same rank (but different from the rank of the pairs, and the triplet)? Say 2D, 2H; 4S, 4C; 6S, 6H; AC, AH, AS; QH, QS, QC, QD is such a hand. (10 points) 6. You are given a deck of 60 cards, which has 10 ranks (from 1 through 10), and 6 suits, called red, blue, white, black, green and yellow. How many 4 card hands are there, in which all the suits and all the ranks are different? Say 2 yellow, 4 green, 5 white, 9 black is such a hand. (12 points) 1
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