Probability HW#1 - Solution Due: March 18, 2010 Problem 1 (2 points). Seven different gifts are to be distributed among 10 children. How many distinct results are possible if no child is to receive more than one gift? Solution. There are 10 × 7! = 604800 7 ways to distribute the gifts. Problem 2 (2 points). If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible? Solution. There are 12 9 5 = 27720 4 5 3 possible divisions. Problem 3 (6 points). We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are 2, 2, 3, and 4 thousand dollars. How many different investment strategies are available if (a) an investment must be made in each opportunity? (b) investments must be made in at least 3 of the 4 opportunities. Solution. (a) Let xi be the number of investments made in i-th opportunity, si be the minimal investment needed toPbe made in the i-th opportunity. That is, s1 = 2, s2 = 2, s3 = 3, s4 = 4, 4i=1 si = 11. Hence we have x1 + x2 + x3 + x4 = 20, 1 where xi ≥ si , 1 ≤ i ≤ 4. Now defining x′i = xi − si for 1 ≤ i ≤ 4, the equation becomes x′1 + x′2 + x′3 + x′4 = 20 − 4 X si = 20 − 11 = 9, i=1 where x′i ≥ 0, 1 ≤ i ≤ 4. The number of different strategies is given by 12 9+4−1 = 220 = 9 4−1 (b) That is, an investment can be made in each opportunity or investments must be made in 3 of the 4 opportunities. The first case is the problem in part (a), which has 220 possible strategies. For the second case, we need to choose 3 of the 4 opportunities. Let i1 , i2 , i3 be the index of the 3 opportunities, and i4 be the index of the rest opportunity. The problem becomes xi1 + xi2 + xi3 = 20 where xij ≥ sij for 1 ≤ j ≤ 4. Defining x′ij = xij − sij for 1 ≤ j ≤ 3. The equation becomes x′i1 + x′i2 + x′i3 = 20 − 4 X sij + si4 = 20 − j=1 4 X si + si4 = 9 + si4 , i=1 where x′ij ≥ 0 for 1 ≤ j ≤ 3. Hence there are 11 + si4 9 + si4 + 3 − 1 = 2 3−1 ways. Then in total we have 4 X 15 14 13 13 11 + si4 = 572 + + + = 220 + 220 + 2 2 2 2 2 i =1 4 choices. Problem 4 (4 points). Give a combinatorial argument (no computations are needed) to establist the following identity: X n i−1 n , = k−1 k i=k where n ≥ k. Hint: Consider the set of numbers 1 through n. How many subsets of size k have i as their highest-numbered member? 2 Solution. Consider the set of numbers 1 through n. The number of subsets of size k is nk . The number of subsets of size k have k as their highest k−1 numbered member is k−1 , since we need to choose k − 1 elements from 1 to k − 1, and add the element k. Similarly, The number of subset of size k i−1 have i (i ≥ k) as their highest-numbered member is k−1 . Hence X n i−1 n = k−1 k i=k Problem 5 (5 points). In how many ways can n identical balls be distributed into r urns so that P the i-th urn contains at least mi balls, for each i = 1, . . . , r? Assume that n ≥ ri=1 mi . Solution. Let xi be the number of balls be distributed into i-th urn, 1 ≤ i ≤ r. Then we have x1 + x2 + · · · + xr = n, where xi ≥ mi . Now defining x′i = xi − mi . Writing the equation as x′1 + x′2 +···+ x′r =n− r X mi , i=1 where x′i ≥ 0. So there’s n− Pr mi + r − 1 r−1 i=1 ways to distribute the balls. Problem 6 (3 points). How many 5-digit numbers can be formed from the integers 1, 2, . . . , 9 if no digit can appear more than twice? (For instance, 41434 is not allowed.) Solution. There are 3 cases. First, the 5-digit number is of the form abcde, we have 9 × 5! = 15120 5 possible numbers. Second, the 5-digit number is of the form aabcd, we have 5! 9 8 · = 30240 1 3 2! possible numbers. Third, the 5-digit number is of the form aabbc, we have 5! 9 7 · = 7560 2 1 2!2! 3 possible numbers. Hence, the total number of solutions is 15120 + 30240 + 7560 = 52920. Problem 7 (4 points). Determine the number of vectors (x1 , . . . , xn ) such that each xi is a positive integer and n X xi ≤ k, i=1 where k ≥ n. Solution. Let x′i = xi − 1 ≥ 0. Then the equation becomes n X x′i = i=1 n X (xi − 1) ≤ k − n, i=1 where x′i ≥ 0. Add appropriate t ≥ 0 to make the equality hold. Hence t+ n X x′i = k − n, i=1 where x′i ≥ 0, t ≥ 0. This gives k k − n + (n + 1) − 1 = n (n + 1) − 1 ways. Problem 8 (6 points). If there are no restrictions on where the digits and letters are placed, how many 8-place license plates consisting of 5 letters and 3 digits are possible if no repetitions of letters or digits are allowed? What if the 3 digits must be consecutive? Solution. If no repetitions of letters or digits are allowed, we have 10 26 · 8! = 318269952000 5 3 ways. If the 3 digits must be consecutive, we have 10 26 · 6! · 3! = 34100352000 5 3 ways. 4