Question

suppose we want to choose 5 objects, without replacement, from 16 distinct objects.

Answer 1

Question:

Suppose we want to choose 5 objects, without replacement, from 16 distinct objects.

A) How many ways can this be done, if the order of the choices is relevant?

B) How many ways can this be done, if the order of the choices is not relevant?

Answer:

A. 4368 ways

B. 524160 ways

Step-by-step explanation:

Given

$Objects = 16$

$Selection = 5$

Required

A & B

Solving (A)

Because the order of choice is irrelevant, this implies combination and it is calculated as follows;

$^nC_r = \frac{n!}{(n-r)!r!}$

Where n = 16 and r = 5

$^{16}C_5 = \frac{16!}{(16-5)!5!}$

$^{16}C_5 = \frac{16!}{11!5!}$

$^{16}C_5 = \frac{16 * 15 * 14 * 13 * 12 * 11!}{11!5!}$

$^{16}C_5 = \frac{16 * 15 * 14 * 13 * 12}{5!}$

$^{16}C_5 = \frac{16 * 15 * 14 * 13 * 12}{5 * 4 * 3 * 2 * 1}$

$^{16}C_5 = \frac{524160}{120}$

$^{16}C_5 = 4368\ ways$

Solving (B)

Because the order of choice is relevant, this implies permutation and it is calculated as follows;

$^nP_r = \frac{n!}{(n-r)!}$

Where n = 16 and r = 5

$^{16}P_5 = \frac{16!}{(16-5)!}$

$^{16}P_5 = \frac{16!}{11!}$

$^{16}P_5 = \frac{16 * 15 * 14 * 13 * 12 * 11!}{11!}$

$^{16}P_5 = 16 * 15 * 14 * 13 * 12$

$^{16}P_5 = 524160\ ways$