Maths Question:
$\text{Verify that}{{\text{ }}^{18}}{{C}_{7}}{{+}^{18}}{{C}_{6}}{{=}^{19}}{{C}_{7}}$
Maths Solution:
$\begin{align} & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}=\frac{18!}{11!7!}+\frac{18!}{12!6!} \\ & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}=\frac{18!}{11!\times 7\times 6!}+\frac{18!}{12\times 11!\times 6!} \\ & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}=\frac{18!}{11!6!}\left( \frac{1}{7}+\frac{1}{12} \right) \\ & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}=\frac{18!}{11!6!}\left( \frac{12+7}{12\times 7} \right) \\ & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}=\frac{19\times 18!}{12\times 11!\times 7\times 6!}=\frac{19!}{12!7!} \\ & ^{18}{{C}_{7}}{{+}^{18}}{{C}_{6}}{{=}^{19}}{{C}_{7}} \\\end{align}$
University mathstopic: