Find the value of ^ 20 C _ 16 - ^ 19 C _ 16 - Vidyadip

Question

Find the value of

${ }^{20} C _{16}-{ }^{19} C _{16}$

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Answer

$\begin{aligned}{ }^{20} C _{16}-{ }^{19} C _{16} & =\frac{20 !}{16 ! 4 !}-\frac{19 !}{16 ! 3 !} \\ & =\frac{20 \times 19 !}{16 ! \times 4 \times 3 !}-\frac{19 !}{16 ! 3 !} \\ & =\frac{19 !}{3 ! 16 !}\left[\frac{20}{4}-1\right] \\ & =\frac{19 !}{3 ! 16 !}(4) \\ & =\frac{19 !}{3 !(16)(15 !)} 4 \\ & =\frac{19 !}{4(3 !)(15 !)} \\ & =\frac{19 !}{4 ! 15 !}={ }^{19} C _{15} \text { or }{ }^{19} C _4\end{aligned}$

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