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 + ..... + n_n = 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 $${2^k} = {2^9}$$

$$ \Rightarrow k = 9$$