If the coefficients of (x^9 ) and (x¹⁰ ) in the binomial expansion of ( (3+ x 2 )^n ) are equal, then (n= )

Step-by-step solution

Given binomial is ( (3+ x 2 )^n ) and the general term in the expansion is ( ^n C_r 3^ n-r 2^r x^r ) ( ) Coefficient of (x^9 ) and (x¹⁰ ) are ( ^n C₉ 3^ n-9 2^9 and ^n C₁₀ 3^ n-10 2¹⁰ respectively ) ( ) It is given that, the coefficients (x^9 ) and (x¹⁰ ) are equal, ( aligned & So, ^n C₉ 3^ n-9 2^9 = ^n C₁₀ 3^ n-10 2¹⁰ & n ! 3 9 !(n-9) ! = n ! 10 !(n-10) ! 2 & 3 n-9 = 1 10 2 n-9=60 & n =69 aligned ) Hence, option (a) is correct.