If the constant term in the expansion of [( 3x^3 - 2x^2 + 5 x^5 )^10] is 2k.l, where l is …

Question

If the constant term in the expansion of

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

Accepted Answer

Note :  Multinomial Theorem :

The general term of $${\left( {{x_1} + {x_2} + ... + {x_n}} \right)^n}$$ the expansion is

$${{n!} \over {{n_1}!{n_2}!...{n_n}!}}x_1^{{n_1}}x_2^{{n_2}}...x_n^{{n_n}}$$

where n1 + n2 + ..... + nn = n

Given,

$${\left( {3{x^2} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$$

$$ = {{{{(3{x^8} - 2{x^7} + 5)}^{10}}} \over {{x^{50}}}}$$

Now constant term in $${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}} = {x^{50}}$$ term in ${(3{x^8} - 2{x^7} + 5)^{10}}$

General term in ${(3{x^8} - 2{x^7} + 5)^{10}}$ is

$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3{x^8})^{{n_1}}}{( - 2{x^7})^{{n_2}}}{(5)^{{n_3}}}$$

$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3)^{{n^1}}}{( - 2)^{{n_2}}}{(5)^{{n^3}}}\,.\,{x^{8{n_1} + 7{n_2}}}$$

$$\therefore$$ Coefficient of ${x^{8{n_1} + 7{n_2}}}$ is

$$ = {{10!} \over {{n_1}!\,{n_2}!\,{n_3}!}}{(3)^{{n_1}}}{( - 2)^{{n_2}}}{(5)^{{n_3}}}$$

where ${n_1} + {n_2} + {n_3} = 0$

For coefficient of x50 :

$8{n_1} + 7{n_2} = 50$

$$\therefore$$ Possible values of n1, n2 and n3 are

n$_1$
    n$_2$
    n$_3$

1
    6
    3

$$\therefore$$ Coefficient of x50

$$ = {{10!} \over {1!\,6!\,3!}}{(3)^1}{( - 2)^6}{(5)^3}$$

$$ = {{10 \times 9 \times 8 \times 7} \over 6} \times 3 \times {5^3} \times {2^6}$$

$$ = 5 \times 3 \times 8 \times 7 \times 3 \times {5^3} \times {2^6}$$

$$ = 7 \times {5^4} \times {3^2} \times {2^9}$$

$$ = {2^k}\,.\,l$$

$$\therefore$$ $$l = 7 \times {5^4} \times {3^2}$$ = An odd integer

and ${2^k} = {2^9}$

$$ \Rightarrow k = 9$$

Solutions · 1