Permutations & Combinations
Exercise 6.2 • Factorial Notation
1. Evaluate (i) 8! (ii) 4! – 3!
(i) 8!
Definition
$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Calculate
$40320$
8! = 40,320
(ii) 4! – 3!
Find 4!
$4 \times 3 \times 2 \times 1 = 24$
Find 3!
$3 \times 2 \times 1 = 6$
Subtract
$24 – 6$
Result = 18
2. Is 3! + 4! = 7!?
LHS
$3! + 4! = 6 + 24 = 30$
RHS
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$
Compare
$30 \neq 5040$
No, 3! + 4! ≠ 7!
3. Compute $\frac{8!}{6! \times 2!}$
Expand
$\frac{8 \times 7 \times 6!}{6! \times 2!}$
Cancel 6!
$\frac{8 \times 7}{2 \times 1}$
Simplify
$\frac{56}{2}$
Answer = 28
4. If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$, find x
Equation
$\frac{1}{6!} + \frac{1}{7 \times 6!} = \frac{x}{8 \times 7 \times 6!}$
Mult 8!
Multiply entire equation by $8!$
Simplify
$\frac{8!}{6!} + \frac{8!}{7!} = x$
Expand
$(8 \times 7) + 8 = x$
Solve
$56 + 8 = x \Rightarrow 64 = x$
x = 64
5. Evaluate $\frac{n!}{(n-r)!}$ when:
(i) n=6, r=2
(ii) n=9, r=5
(i) n=6, r=2
(ii) n=9, r=5
(i) n = 6, r = 2
Substitute
$\frac{6!}{(6-2)!} = \frac{6!}{4!}$
Expand
$\frac{6 \times 5 \times 4!}{4!}$
Result
$6 \times 5 = 30$
Answer = 30
(ii) n = 9, r = 5
Substitute
$\frac{9!}{(9-5)!} = \frac{9!}{4!}$
Expand
$\frac{9 \times 8 \times 7 \times 6 \times 5 \times 4!}{4!}$
Multiply
$9 \times 8 \times 7 \times 6 \times 5$
Result
$15120$
Answer = 15,120