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

The correct option is A.

Explanation:

Combination formula for selecting r items from n items is

`_nC_r=\binom{n}{r}=(n!)/(r!(n-r)!)`

Using the formula we get

`_7C_2=\binom{7}{2}=(7!)/(2!(7-2)!)=(7* 6* 5!)/(2* 1* 5!)=21`

Therefore option A is correct.

`_(21)C_2=\binom{21}{2}=(21!)/(2!(21-2)!)=(21* 20* 19!)/(2* 1* 19!)=210`

Therefore option B is incorrect.

`_(21)C_7=\binom{21}{7}=(21!)/(7!(21-7)!)=116280`

Therefore option C is incorrect.

`_(7)C_7=\binom{7}{7}=(7!)/(7!(7-7)!)=1`

Therefore option D is incorrect.

Answer 2

Pls. see attachment.