Binomial theorem and factorials. Game on.

$C(5,2) = \dfrac{5!}{2! \cdot 3!} = \dfrac{120}{2 \cdot 6} = 10$.

$4! = 4 \times 3 \times 2 \times 1 = 24$.

$C(4,1) = \dfrac{4!}{1! \cdot 3!} = \dfrac{24}{1 \cdot 6} = 4$.

$3! = 3 \times 2 \times 1 = 6$.

$C(5,3) = \dfrac{5!}{3!\,(5-3)!} = 10$.

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.

$C(6,3) = \dfrac{6!}{3!\,(6-3)!} = 20$.

$2! = 2 \times 1 = 2$.