The natural number m, for which the coefficient of x in the binomial expansion of
${\left( {{x^m} + {1 \over {{x^2}}}} \right)^{22}}$ is 1540, is .............
Answer (integer)
13
Solution
General term,
<br><br>$${T_{r + 1}} = {}^{22}{C_r}{({x^m})^{22 - r}}{\left( {{1 \over {{x^2}}}} \right)^r} = {}^{22}{C_r}{x^{22m - mr - 2r}}$$<br><br>$\because$ ${}^{22}{C_3} = {}^{22}{C_{19}} = 1540$<br><br>$\therefore$ $r = 3\,or\,19$<br><br>$22m - mr - 2r = 1$<br><br>$m = {{2r + 1} \over {22 - 5}}$<br><br>When $r = 3$, $m = {7 \over {19}} \notin N$<br><br>When $r = 19$, $m = {{38 + 1} \over {22 - 19}} = {{39} \over 3} = 13$<br><br>$\therefore$ $m = 13$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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