If the constant term in the expansion of

${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$ is 2^k.l, where l is an odd integer, then the value of k is equal to:

<b>Note : </b> <b>Multinomial Theorem : </b> <br><br>The general term of ${\left( {{x_1} + {x_2} + ... + {x_n}} \right)^n}$ the expansion is <br><br>${{n!} \over {{n_1}!{n_2}!...{n_n}!}}x_1^{{n_1}}x_2^{{n_2}}...x_n^{{n_n}}$ <br><br>where n₁ + n₂ + ..... + n_n = n <br/><br/><p>Given,</p> <p>${\left( {3{x^2} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$</p> <p>$= {{{{(3{x^8} - 2{x^7} + 5)}^{10}}} \over {{x^{50}}}}$</p> <p>Now constant term in ${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}} = {x^{50}}$ term in ${(3{x^8} - 2{x^7} + 5)^{10}}$</p> <p>General term in ${(3{x^8} - 2{x^7} + 5)^{10}}$ is</p> <p>$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3{x^8})^{{n_1}}}{( - 2{x^7})^{{n_2}}}{(5)^{{n_3}}}$$</p> <p>$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3)^{{n^1}}}{( - 2)^{{n_2}}}{(5)^{{n^3}}}\,.\,{x^{8{n_1} + 7{n_2}}}$$</p> <p>$\therefore$ Coefficient of ${x^{8{n_1} + 7{n_2}}}$ is</p> <p>$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3)^{{n_1}}}{( - 2)^{{n_2}}}{(5)^{{n_3}}}$$</p> <p>where ${n_1} + {n_2} + {n_3} = 0$</p> <p>For coefficient of x⁵⁰ :</p> <p>$8{n_1} + 7{n_2} = 50$</p> <p>$\therefore$ Possible values of n₁, n₂ and n₃ are</p> <p><style type="text/css"> .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-baqh{text-align:center;vertical-align:top} </style> <table class="tg" style="undefined;table-layout: fixed; width: 253px"> <colgroup> <col style="width: 75px"> <col style="width: 85px"> <col style="width: 93px"> </colgroup> <thead> <tr> <th class="tg-baqh">n$_1$</th> <th class="tg-baqh">n$_2$</th> <th class="tg-baqh">n$_3$</th> </tr> </thead> <tbody> <tr> <td class="tg-baqh">1</td> <td class="tg-baqh">6</td> <td class="tg-baqh">3</td> </tr> </tbody> </table></p> <p>$\therefore$ Coefficient of x⁵⁰</p> <p>$= {{10!} \over {1!\,6!\,3!}}{(3)^1}{( - 2)^6}{(5)^3}$</p> <p>$= {{10 \times 9 \times 8 \times 7} \over 6} \times 3 \times {5^3} \times {2^6}$</p> <p>$= 5 \times 3 \times 8 \times 7 \times 3 \times {5^3} \times {2^6}$</p> <p>$= 7 \times {5^4} \times {3^2} \times {2^9}$</p> <p>$= {2^k}\,.\,l$</p> <p>$\therefore$ $l = 7 \times {5^4} \times {3^2}$ = An odd integer</p> <p>and ${2^k} = {2^9}$</p> <p>$\Rightarrow k = 9$</p>