Question

prove that 2*6*10*14....upto n factors = (2n)!/n! = (n+1)(n+2)....to n factors
(from chapter: permutation and combination

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Answer 1

✪PROOF✪

FIRST PART

$2 \times 6 \times 10 \times 14...n \: factors$

$=2.6.10.14.18.....[2(2n - 1)]$

$=(2.1)(2.3)(2.5)......[2(2n - 1)]$

$={2}^{n}[1.3.5.7.9.........(2n - 1)]$

$=\frac{ {2}^{n} [1.3.5...(2n - 1)][1.2.3.4......n]}{1.2.3.4.....n}$

$= \frac{[1.3.5.7...(2n - 1)][2.4.6.8......(2n)]}{1.2.3.4.5.....n}$

$= \frac{1.2.3.4.5.6.7.8........(2n - 1)(2n)}{1.2.3.4.........n}$

$= \frac{(2n)!}{n!}$

SECOND PART

$\frac{(2n)!}{n!}$

$= \frac{1.2.3.4..(n - 1)n(n + 1)(n + 2)...(2n-1)(2n)}{1.2.3.4......(n - 1)n}$

$= (n + 1)(n + 2)......(2n - 1)(2n)$

Hence Proved