Discrete Mathematics with Graph Theory (3rd Edition) 191

Discrete Mathematics with Graph Theory (3rd Edition) 191 -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 7.3 189 ( n -1) (n -1) 25. (a) k _ 1 + k (n-1)! + (n-1)! [(n - 1) - (k -l)]!(k -1)! (n -1- k)!k! = (n - 1)! + (n - 1)! (n - k)!(k - 1)! (n - 1 - k)!k! (n - l)!k + (n - l)!(n - k) (n - k)!k! (n -l)![k + (n - k)] = (n -l)!n = n! = (n) (n - k)!k! (n - k)!k! (n - k)!k! k (b) Suppose you want to select r people from a group of n men and one woman. You can either select r men, or r - 1 men and the woman. Now the identity follows from the addition rule. 26. (a) (2;) = 2n(2;-1) = n(2n-1) = 2n 2 -n; 2(;)+n2 = 2 n (n 2 -1) +n 2 = n(n-1)+n2 = 2n 2 -n. (b) (2;) is the number of ways of choosing 2 objects from 2n. From a different point of view, however, suppose we divide the objects into two sets of n objects each. To choose 2 objects we can choose from the first set (in (;) ways), from the second set (also in (;) ways) or one from each set (in n(n) ways). So the number of ways of choosing 2 objects from 2n is (;) + (;) +n 2 = 2(;) +n 2 . Exercises 7.3 1. [BB] (a)
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (b) ]6 = l2 (c) Since the possibilities are (1,2), (2,3), (3,4), (4,5), (5,6) and their reverses, the answer is ~~ = 5 18 (d) There are 3 x 3 = 9 cases where the odd number is first and 9 more where the even number is fi Th . 18 1 rst. e answer IS 36 = "2. 2. (a)~; (b) ~ -1 6 -3' (e) Since the only possible number is 3, the answer is i. 3. (a) 3~ (b) The complement of this event is {(5, 6), (6,5), (6, 6)}, so the answer is 1-33 6 = g. (c) 70 = 2 x 5 x 7. The possibilities are (1, 1), (1,2), (1,5), (2,5) and their reverses. The answer is 3~. (d) The possible sums are 5 and 10. The answer is W = ;6. 4. (a) The only possibility is (5,5,5), so the answer is 2~6. (b) The complement of the event is {(6, 6, 5), (6,5,6), (5,6,6), (6,6, 6)}, so the answer is 1-2f6 = 53 54 (c) The prime numbers which can appear are 2, 3, 5, and these can be arranged in 3! ways. Th . 6 1 e answer IS 216 = 36 (d) ~~~~: = ~....
View Full Document

Ask a homework question - tutors are online