Question

prove that 15c2 = 105​

Answer 1

$\large \underline \bold{Solution :-}$

$\small \bold{15 \displaystyle {c} \displaystyle {2}}$

$\large \underline \bold{We \: know \: that -}$

$\small \bold{\sf\boxed{n \displaystyle {c} \displaystyle{r} =\dfrac{n!}{r! (n-r)!}}}$

$\small \bold{Here \: ,}$

$\: \: \:$$\small \bold{n = 15}$

$\: \: \:$$\small \bold{r = 2}$

So ,

$\small \bold{=\dfrac{15!}{2! (15-2)!}}$

$\small \bold{=\dfrac{15!}{2! \: 13!}}$

$\large \underline \bold{We \: know \: that -}$

$\large \bold{\sf\boxed{n! = n(n - 1)(n - 2)....3.2.1}}$

So ,

$\small \bold{\dfrac{=15\times 14\times \cancel{13!}}{2\times \cancel{13!}}}$

$\small \bold{=\dfrac{15\times 14}{2}}$

$\small \bold{=15\times 7}$

$\small \bold{=105}$

$\large \bold{HENCE \: PROOF}$

$\small \underline \bold{Extra \: Usable \: Identity -}$

★ $\small \bold{For \: combination -}$

$\small \bold{If \: , \: ncx = ncy}$

$\small \bold{then \: , \: n = x + y}$

Answer 2

Answer:

$~≥2}}15c2$