The natural number m, for which the coefficient of x in the binomial expansion of ( x^m + 1 / x^2 )^22 is 1540, is .............

General term, T_r + 1 = ^22C_r(x^m)^22 - r( 1 / x^2 )^r = ^22C_rx^22m - mr - 2r Because ^22C_3 = ^22C_19 = 1540 Therefore r = 3 or 19 22m - mr - 2r = 1 m = 2r + 1 / 22 - 5 When r = 3, m = 7 / 19 notin N When r = 19, m = 38 + 1 / 22 - 19 = 39 / 3 = 13 Therefore m = 13

The natural number m for which the coefficient of x in ... | null (Mathematics)

General term,

{T_{r + 1}} = {}^{22}{C_r}{({x^m})^{22 - r}}{\left( {{1 \over {{x^2}}}} \right)^r} = {}^{22}{C_r}{x^{22m - mr - 2r}}

\because {}^{22}{C_3} = {}^{22}{C_{19}} = 1540

\therefore r = 3\,or\,19

22m - mr - 2r = 1

m = {{2r + 1} \over {22 - 5}}

When

r = 3,

m = {7 \over {19}} \notin N

When

r = 19,

m = {{38 + 1} \over {22 - 19}} = {{39} \over 3} = 13

\therefore m = 13

Explanation

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