Question

if 13C6 +13C5+13C5+13C4 = 15Cx then X=​

Answer 1

For this, we use the concept,

$\large^n\!C_r\ +\ ^n\!C_{r+1}\ =\ ^{n+1}\!C_{r+1}$

And,

$\large^n\!C_r=\ ^n\!C_{n-r}$

So,

$\begin{aligned}&^{13}\!C_6\ +\ ^{13}\!C_5\ +\ ^{13}\!C_5\ +\ ^{13}\!C_4\ =\ ^{15}\!C_x\\\\\implies\ \ &\left(^{13}\!C_4\ +\ ^{13}\!C_5\right)\ +\ \left(^{13}\!C_5\ +\ ^{13}\!C_6\right)\ =\ ^{15}\!C_x\\\\\implies\ \ &^{14}C_5+\ ^{14}C_6=\ ^{15}C_x\\\\\implies\ \ &^{15}C_6=\ ^{15}C_x\\\\\implies\ \ &\Large\boxed{x=6\quad\text{OR}\quad x=9}\end{aligned}$