CHAP. 5] TECHNIQUES OF COUNTING 103
SupplementaryProblems
FACTORIAL NOTATION, BINOMIAL COEFFICIENTS
5.31. Find: (a) 10!, 11!, 12!; (b) 60!. (Hint: Use Sterling’s approximation ton!.)
5.32. Evaluate: (a) 16!/14!; (b) 14!/11!; (c) 8!/10!; (d) 10!/13!.5.33. Simplify: (a)(n+ 1 )!
n!
; (b)n!
(n− 2 )!
; (c)(n− 1 )!
(n+ 2 )!
; (d)(n−r+ 1 )!
(n−r− 1 )!
.5.34. Find: (a)(
5
2)
; (b)(
7
3)
; (c)(
14
2)
; (d)(
6
4)
; (e)(
20
17)
; (f)(
18
15)
.5.35. Show that: (a)(
n
0)
+(
n
n)
+(
n
2)
+(
n
3)
+···+(
n
n)
= 2 n(b)(
n
0)
−(
n
1)
+(
n
2)
−(
n
3)
+···+(
n
n)
= 05.36. Given the following eighth row of Pascal’s triangle, find: (a) the ninth row; (b) the tenth row.182856705628815.37. Evaluate the following multinomial coefficients (defined in Problem 5.29):(a)(
6
2 , 3 , 1)
; (b)(
7
3 , 2 , 2 , 0)
; (c)(
9
3 , 5 , 1)
; (d)(
8
4 , 3 , 2)
.COUNTING PRINCIPLES
5.38. A store sells clothes for men. It has 3 kinds of jackets, 7 kinds of shirts, and 5 kinds of pants. Find the number of
ways a person can buy: (a) one of the items; (b) one of each of the three kinds of clothes.
5.39. A class has 10 male students and 8 female students. Find the number of ways the class can elect: (a) a class
representative; (b) 2 class representatives, one male and one female; (c) a class president and vicepresident.
5.40. Suppose a code consists of five characters, two letters followed by three digits. Find the number of: (a) codes;
(b) codes with distinct letter; (c) codes with the same letters.PERMUTATIONS
5.41. Find the number of automobile license plates where: (a) Each plate contains 2 different letters followed by 3 different
digits. (b) The first digit cannot be 0.
5.42. Find the numbermof ways a judge can award first, second, and third places in a contest with 18 contestants.
5.43. Find the number of ways 5 large books, 4 medium-size books, and 3 small books can be placed on a shelf where:
(a) there are no restrictions; (b) all books of the same size are together.
5.44. A debating team consists of 3 boys and 3 girls. Find the number of ways they can sit in a row where:
(a) there are no restrictions; (b) the boys and girls are each to sit together; (c) just the girls are to sit together.
5.45. Find the number of ways 5 people can sit in a row where: (a) there are no restrictions; (b) two of the people insist on
sitting next to each other.
5.46. Repeat Problem 5.45 if they sit around a circular table.
5.47. Consider all positive integers with three different digits. (Note that zero cannot be the first digit.) Find the number of
them which are: (a) greater than 700; (b) odd; (c) divisible by 5.
5.48. Suppose repetitions are not permitted. (a) Find the number of three-digit numbers that can be formed from the six
digits 2, 3, 5, 6, 7, and 9. (b) How many of them are less than 400? (c) How many of them are even?
5.49. Find the numbermof ways in which 6 people can ride a toboggan if one of 3 of them must drive.
5.50. Findnif: (a)P (n, 4 )= 42 P (n, 2 ); (b) 2P (n, 2 )+ 50 =P( 2 n, 2 ).