CHAP. 5] TECHNIQUES OF COUNTING 97
(b)
7!
10!
=
7!
10 · 9 · 8 · 7!
=
1
10 · 9 · 8
=
1
720
.5.3. Simplify: (a)
n!
(n− 1 )!;(b)(n+ 2 )!
n!.(a)
n!
(n− 1 )!
=
n(n− 1 )(n− 2 )··· 3 · 2 · 1
(n− 1 )(n− 2 )··· 3 · 2 · 1
=n; alternatively,
n!
(n− 1 )!
=
n(n− 1 )!
(n− 1 )!
=n.(b)(n+ 2 )!
n!
=(n+ 2 )(n+ 1 )n!
n!
=(n+ 2 )(n+ 1 )=n^2 + 3 n+2.5.4. Compute: (a)
(
16
3)
;(b)(
12
4)
;(c)(
8
5)
.Recall that there are as many factors in the numerator as in the denominator.(a)(
16
3)
=16 · 15 · 14
3 · 2 · 1
=560; (b)(
12
4)
=12 · 11 · 10 · 9
4 · 3 · 2 · 1
=495;(c) Since 8− 5 =3, we have(
8
5)
=(
8
3)
=
8 · 7. 6
3 · 2 · 1
=56.5.5. Prove:
(
17
6)
=(
16
5)
+(
16
6)
.Now(
16
5)
+(
16
6)
=
16!
5! 11!
+
16!
6! 10!. Multiply the first fraction by^66 and the second by^1111 to obtain the same
denominator in both fractions; and then add:
(
16
5
)
+(
16
6)
=
6 · 16!
6 · 5 !· 11!
+
11 · 16!
6 !· 11 · 10!
=
6 · 16!
6 !· 11!
+
11 · 16!
6 !· 11!=
6 · 16 !+ 11 · 16!
6 !· 11!=
( 6 + 11 )· 16!
6 !· 11!=
17 · 16!
6 !· 11!=
17!
6 !· 11!=(
17
6)5.6. Prove Theorem 5.3:
(
n+ 1
r)
=(
n
r− 1)
+(
n
r)
.(The technique in this proof is similar to that of the preceding problem.)Now(
n
r− 1)
+(
n
r)
=
n!
(r− 1 )!·(n−r+ 1 )!+
n!
r!·(n−r)!.To obtain the same denominator in both fractions, multiply the first fraction byrrand the second fraction by
n−r+ 1
n−r+ 1.
Hence (
n
r− 1)
+(
n
r)
=r·n!
r·(r− 1 )!·(n−r+ 1 )!
+(n−r+ 1 )·n!
r!·(n−r+ 1 )·(n−r)!=
r·n!
r!(n−r+ 1 )!+
(n−r+ 1 )·n!
r!(n−r+ 1 )!=
r·n!+(n−r+ 1 )·n!
r!(n−r+ 1 )!=
[r+(n−r+ 1 )]·n!
r!(n−r+ 1 )!=(n+ 1 )n!
r!(n−r+ 1 )!
=(n+ 1 )!
r!(n−r+ 1 )!
=(
n+ 1
r)COUNTING PRINCIPLES
5.7. Suppose a bookcase shelf has 5 History texts, 3 Sociology texts, 6 Anthropology texts, and 4 Psychology
texts. Find the numbernof ways a student can choose:
(a) one of the texts; (b) one of each type of text.
(a) Here the Sum Rule applies; hence,n= 5 + 3 + 6 + 4 =18.
(b) Here the Product Rule applies; hence,n= 5 · 3 · 6 · 4 =360.