7. binomial theoram — ગણિત ધોરણ 11 સાયન્સ — Question

$=(1+63)^{n}-(63-1)^{2 n+1}$ 

$=(1+63)^{n}+(1-63)^{2 n+1}$

$=\left(1+^{n} C_{1} 63+^{n} C_{2}(63)^{2}+\ldots+(63)^{n}\right.$

$+\left[1-^{(2 n+1)} C_{1} 63+(2 n+1) C_{2}(63)^{2}\right.$

$\left.+\ldots+(-1)(63)^{(2 n+1)}\right] $

$=2+63\left[^{n} C_{1}+^{n} C_{2}(63)+\ldots+(63)^{n-1}\right] $

$\left.+^{(2 n+1)} C_{2}(63)-\ldots-(63)^{(2 n)}\right]$

Hence, the remainder is $2.$