Permutation and Combination Maharashtra Board EXERCISE 3.2

EXERCISE 3.2

1. Evaluate:

(i) 8!

=8×7×6×5×4×3×2×1

=40320

(ii)10!

=10×9×8×7×6×5×4×3×2×1

=3628800

(iii)10!-6!

=10×9×8×7×6!-6!

=6!(10×9×8×7-1)

=3628080

(iv) (10-6)!

=4!

=4×3×2×1

=24

2. Compute

(i) $\frac{12!}{6!}$ $\frac{\text{12!}}{\text{6!}}$

\frac{12!}{6!}

$\frac{\text{12!}}{\text{6!}}$

\frac{12x11x10x9x8x7x6!}{6!}

$\frac{\text{12x11x10x9x8x7x6!}}{\text{6!}}$

=12×11×10×9×8×7

=665280

(ii) $(\frac{12}{6})!$ $\left(\frac{\text{12}}{\text{6}}\right)!$

(\frac{12}{6})!

$\left(\frac{\text{12}}{\text{6}}\right)!$

= 2!

(iii) (3×2)!

= 6!

=6×5×4×3×2×1

=720

(iv) 3!×2!

= 3×2×1×2×1

=12

(v) $\frac{9!}{3! 6!}$ $\frac{\text{9!}}{\text{3! 6!}}$

\frac{9 8 7 6!}{3! 6!}

$\frac{\text{9 8 7 6!}}{\text{3! 6!}}$

\frac{504}{6}

$\frac{\text{504}}{\text{6}}$

=84

(vi) $\frac{6!-4!}{4!}$ $\frac{\text{6!-4!}}{\text{4!}}$

\frac{6×5×4!-4!}{4!}

$\frac{\text{6×5×4!-4!}}{\text{4!}}$

4! (\frac{6×5-1}{4!})

$4!\left(\frac{\text{6×5-1}}{\text{4!}}\right)$

=30-1

=29

(vii) $\frac{8!}{6!-4!}$ $\frac{\text{8!}}{\text{6!-4!}}$

\frac{8×7×6×5×4!}{6×5×4!-4!}

$\frac{\text{8×7×6×5×4!}}{\text{6×5×4!-4!}}$

\frac{4!}{4!} (\frac{8×7×6×5}{6×5-1})

$\frac{\text{4!}}{\text{4!}}\left(\frac{\text{8×7×6×5}}{\text{6×5-1}}\right)$

\frac{1680}{30-1}

$\frac{\text{1680}}{\text{30-1}}$

=57.93

(viii) $\frac{8!}{(6-4)!}$ $\frac{\text{8!}}{\text{(6-4)!}}$

\frac{8!}{(6-4)!}

$\frac{\text{8!}}{\text{(6-4)!}}$

\frac{8×7×6×5×4×3×2!}{2!}

$\frac{\text{8×7×6×5×4×3×2!}}{\text{2!}}$

=40320

3. Write in terms of factorials

(i)5×6×7×8×9×10

= \frac{10×9×8×7×6×5×4×3×2×1}{4×3×2×1}

$=\frac{\text{10×9×8×7×6×5×4×3×2×1}}{\text{4×3×2×1}}$

= \frac{10!}{4!}

$=\frac{\text{10!}}{\text{4!}}$

(ii)3×6×9×12×15

=1×3×2×3×3×3×4×3×5×3

=1×2×3×4×5×3×3×3×3×3

= 5! \times 3^{5}

$=5!×3^{5}$

(iii)6×7×8×9

= \frac{9×8×7×6×5×4×3×2×1}{5×4×3×2×1}

$=\frac{\text{9×8×7×6×5×4×3×2×1}}{\text{5×4×3×2×1}}$

= \frac{9!}{5!}

$=\frac{\text{9!}}{\text{5!}}$

(iv) 5×10×15×20

=1×5×2×5×3×5×4×5

= 4! \times 5^{4}

$=4!×5^{4}$

4. Evaluate

$\frac{n!}{r!(n-r)!}$ for

(i) n=8, r=6 (i) n=12, r=12 (i) n=15, r=10 (i) n=15, r=8

solution:

(i)

\frac{8!}{6!(8-6)!}

$\frac{\text{8!}}{\text{6!(8-6)!}}$

\frac{8×7×6!}{6!2!}

$\frac{\text{8×7×6!}}{text{6!2!}}$

= \frac{56}{2} = 28

$=\frac{56}{2}=28$

(ii)

\frac{12!}{12!(12-12)!}

$\frac{\text{12!}}{\text{12!(12-12)!}}$

\frac{1}{0!}

$\frac{\text{1}}{\text{0!}}$

(iii)

\frac{15!}{10!(15-10)!}

$\frac{\text{15!}}{\text{10!(15-10)!}}$

\frac{151413121110!}{10!5!}

$\frac{\text{151413121110!}}{\text{10!5!}}$

\frac{360360}{120}

$\frac{\text{360360}}\{\text{120}}$

=3003

(iv)

\frac{15!}{8!(15-8)!}

$\frac{\text{15!}}{\text{8!(15-8)!}}$

\frac{15×14×13×12×11×10×9×8!}{8!5!}

$\frac{\text{15×14×13×12×11×10×9×8!}}{\text{8!5!}}$

\frac{32432400}{120}

$\frac{\text{32432400}}\{\text{120}}$

=270270

5. Find n, if

(i) $\frac{n}{8!} = \frac{3}{6!} + \frac{1!}{2!}$ $\frac{\text{n}}{\text{8!}}=\frac{\text{3}}{\text{6!}}+\frac{\text{1!}}{\text{2!}}$

\frac{n}{8!} = \frac{3 4!+6!}{! 4!}

$\frac{\text{n}}{\text{8!}}=\frac{\text{3 4!+6!}}{\text6!4!}$

\frac{n}{8!} = 4! \frac{3+6 5}{6!4!}

$\frac{\text{n}}{\text{8!}}=4!\frac{\text{3+6 5}}{\text{6!4!}$

\frac{n}{8!} = \frac{3+30}{6!}

$\frac{\text{n}}{\text{8!}}=\frac{\text{3+30}}{\text{6!}}$

\frac{n}{8!} = \frac{33}{6!}

$\frac{\text{n}}{\text{8!}}=\frac{\text{33}}{\text{6!}}$

n = \frac{33×8×7×6!}{6!}

$n=\frac{\text{33×8×7×6!}}{\text{6!}}$

n = 1848

$n=1848$

(ii) $\frac{n}{6!} = \frac{4}{8!} + \frac{3}{6!}$ $\frac{\text{n}}{\text{6!}}=\frac{\text{4}}{\text{8!}}+\frac{\text{3}}{\text{6!}}$

\frac{n}{6!} = \frac{4 6!+3×8!}{8!6!}

$\frac{\text{n}}{\text{6!}}=\frac{\text{4 6!+3×8!}}{\text{8!6!}}$

\frac{n}{6!} = \frac{6!(4+3×8×7)}{8!6!}

$\frac{\text{n}}{\text{6!}}=\frac{\text{6!(4+3×8×7)}}{\text{8!6!}}$

n = \frac{4+3×8×7}{8×7}

$n=\frac{\text{4+3×8×7}}{\text{8×7}}$

= \frac{4(1+3×8×7)}{8×7}

$=\frac{\text{4(1+3×8×7)}}{\text{8×7}}$

= \frac{1+42}{14}

$=\frac{\text{1+42}}{\text{14}}$

= \frac{43}{14}

$=\frac{\text{43}}{\text{14}}$