Question

Which of the following demonstrates how the first 21 (on the left side of the triangle) is calculated using the combination pattern?

Answer 1

Pls. see attachment. 

Answer 2

Answer:

The correct option is A.

Step-by-step explanation:

Combination formula for selecting r items from n items is

$_nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}$

Using the formula we get

$_7C_2=\binom{7}{2}=\frac{7!}{2!(7-2)!}=\frac{7\times 6\times 5!}{2\times 1\times 5!}=21$

Therefore option A is correct.

$_{21}C_2=\binom{21}{2}=\frac{21!}{2!(21-2)!}=\frac{21\times 20\times 19!}{2\times 1\times 19!}=210$

Therefore option B is incorrect.

$_{21}C_7=\binom{21}{7}=\frac{21!}{7!(21-7)!}=116280$

Therefore option C is incorrect.

$_{7}C_7=\binom{7}{7}=\frac{7!}{7!(7-7)!}=1$

Therefore option D is incorrect.