Question

The value of ${}^{15}{C}_8{+}^{15}{C}_9{-}^{15}{C}_6{-}^{15}{C}_7$ is

Answer 1

Finding the sum of series
It is given that ${(}^{15}{C}_8{+}^{15}{C}_9)-{(}^{15}{C}_6{+}^{15}{C}_7)$
Using ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$
${=}^{16}{C}_9{-}^{16}{C}_7$
${=}^{16}{C}_9{-}^{16}{C}_{16-7}$
${=}^{16}{C}_9{-}^{16}{C}_9=0$
Hence, the value of
${}^{15}{C}_8{+}^{15}{C}_9{-}^{15}{C}_6{-}^{15}{C}_7$ is 0.

​​