Simplify: $^{(n + 1)} C_{3} -^{(n - 1)} C_{3} =$

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Solution

$^{n + 1} C_{3} -^{n - 1} C_{3}$
$\frac{(n + 1)!}{(n - 2)! 3!} - \frac{(n - 1)!}{(n - 4) 3!}$ as $[^{n} C_{r} = \frac{n!}{(n - 7)! r!}]$
$\frac{1}{3!} [(n + 1) (n) (n - 1) - (n - 1) (n - 2) (n - 3)]$
$\frac{(n - 1)}{3!} [n^{2} + n - (n - 2) (n - 3)]$
$\frac{(n - 1)}{6} (n^{2} + n - (n^{2} - 2 n - 3 n + 6))$
$\frac{(n - 1)}{6} [n^{2} + n - n^{2} + 5 n - 6]$
$\frac{(n - 1)}{6} \times (6 n - 6) = \frac{6 (n - 1)^{2}}{6} = (n - 1)^{2}$

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^{n - 1} C_{3} +^{n - 1} C_{4} >^{n} C_{3}

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^{2 n} C_{3} :^{n} C_{3} = 12 : 1

^{(n + 1)} C_{1} +^{(n + 1)} C_{2} +^{(n + 1)} C_{3} + \dots . . +^{(n + 1)} C_{n} =

^{n + 1} C_{3} = 4^{n} C_{2}

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^{n} C_{3} :^{n - 1} C_{4} = 20 : 21

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