Given Binomial expression is

$${\left( {2{x^3} + {3 \over {{x^k}}}} \right)^{12}}$$

General term,

$${T_{r + 1}} = {}^{12}{C_r}{(2{x^3})^r}\,.\,{\left( {{3 \over {{x^k}}}} \right)^{12 - r}}$$

$$ = \left( {{}^{12}{C_r}\,.\,{2^r}\,.\,{3^{12 - r}}} \right)\,.\,{x^{3r - 12k + kr}}$$

For constant term,

$$3r - 12k + kr = 0$$

$$ \Rightarrow k(12 - r) = 3r$$

$$ \Rightarrow k = {{3r} \over {12 - r}}$$

For r = 1, $$k = {3 \over {11}}$$ (not integer)

For r = 2, $$k = {6 \over {10}}$$ (not integer)

For r = 3, $$k = {9 \over {9}}=1$$ (integer)

For r = 6, $$k = {18 \over {6}}=3$$ (integer)

For r = 8, $$k = {24 \over {4}}=6$$ (integer)

For r = 9, $$k = {27 \over {3}}=9$$ (integer)

For r = 10, $$k = {30 \over {2}}=15$$ (integer)

For r = 11, $$k = {33 \over {1}}=33$$ (integer)

So, for r = 3, 6, 8, 9, 10 and 11 k is positive integer.

When k = 1 then r = 3 and constant term is

$$ = {}^{12}{C_3}\,.\,{2^3}\,.\,{3^9}$$

$$ = {{12\,.\,11\,.\,10} \over {3\,.\,2\,.\,1}}\,.\,{2^3}\,.\,{3^9}$$

$$ = 2\,.\,11\,.\,2\,.\,5\,.\,{2^3}\,.\,{3^9}$$

$$ = 11\,.\,5\,.\,{2^5}\,.\,{3^9}$$

$$ = {2^5}\,.\,(55\,.\,{3^9})$$

$$ = {2^5}(l)$$

$$ \ne {2^8}\,.\,l$$

When x = 3 then r = 6 and constant term

$$ = {}^{12}{C_6}\,.\,{2^6}\,.\,{3^6}$$

$$ = {{12\,.\,11\,.\,10\,.\,9\,.\,8\,.\,7} \over {6\,.\,5\,.\,4\,.\,3\,.\,2\,.\,1}}\,.\,{2^6}\,.\,{3^6}$$

$$ = {2^8}\,.\,231\,.\,{3^6}$$

$$ = {2^8}(l)$$

When k = 6 then r = 8 and constant term

$$ = {}^{12}{C_8}\,.\,{2^8}\,.\,{3^4}$$

$$ = {{12\,.\,11\,.\,10\,.\,9} \over {4\,.\,3\,.\,2\,.\,1}}\,.\,{2^8}\,.\,{3^4}$$

$$ = {2^8}\,.\,55\,.\,{3^6}$$

$$ = {2^8}(l)$$

When x = 9 then r = 9 and constant term

$$ = {}^{12}{C_9}\,.\,{2^9}\,.\,{3^3}$$

$$ = {{12\,.\,11\,.\,10} \over {3\,.\,2\,.\,1}}\,.\,{2^9}\,.\,{3^3}$$

$$ = {2^{11}}\,.\,55\,.\,{3^3}$$

Here power of 2 is 11 which is greater than 8. So, k = 9 is not possible.

Similarly for k = 15 and k = 33, $${2^8}\,.\,l$$ form is not possible.

$$\therefore$$ k = 3 and k = 6 is accepted.

$$\therefore$$ For 2 positive integer value of k, $${2^8}\,.\,l$$ form of constant term possible.

Binomial Theorem

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