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The coefficient of \[{x^{24}}\] in the expansion of ${(1 + {x^2})^{12}}\,(1 + {x^{12}})\,(1 + {x^{24}})$ is
A) $^{12}{C_6}$
B) $^{12}{C_6} + 2$
C) $^{12}{C_6} + 4$
D) $^{12}{C_6} + 6$

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Last updated date: 20th Jun 2024
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Answer
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Hint: We can see that in this question ${(1 + {x^2})^{12}}$ is in the form of ${(x + y)^n}$ where $x = 1$ and $y={x^2}$. We also know that the binomial expansion of above expression is ${(x + y)^n}{ = ^n}{C_0}{ + ^n}{C_1}{x^1}{ + ^n}{C_2}{x^2}{ + ^n}{C_3}{x^3} + ....{\,^n}{C_n}{x^n}$. We will expand the above expression in this form. Then, we will multiply $(1 + {x^{12}})\,(1 + {x^{24}})$ these expressions together. Then we will try to find the coefficient of ${x^{24}}$.

Formula used:
${(x + y)^n}{ = ^n}{C_0}{ + ^n}{C_1}{x^1}{ + ^n}{C_2}{x^2}{ + ^n}{C_3}{x^3} + ....{\,^n}{C_n}{x^n}$and $^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$

Complete step by step answer:
We know that Binomial expansion of ${(x + y)^n}{ = ^n}{C_0}{ + ^n}{C_1}{x^1}{ + ^n}{C_2}{x^2}{ + ^n}{C_3}{x^3} + ....{\,^n}{C_n}{x^n}$
We have ${(1 + {x^2})^{12}}\,(1 + {x^{12}})\,(1 + {x^{24}})$
We will expand ${(1 + {x^2})^{12}}$ by using Binomial expansion as above and we will multiply these two expressions together $(1 + {x^{12}})\,(1 + {x^{24}})$.
Thus, ${(^{12}}{C_0}{ + ^{12}}{C_1}{x^2}{ + ^{12}}{C_2}{x^4}{ + ^{12}}{C_3}{x^6}{ + ^{12}}{C_4}{x^8}{ + ^{12}}{C_5}{x^{10}}{ + ^{12}}{C_6}{x^{12}} + {.....^{12}}{C_{12}}{x^{24}})\,(1 + {x^{12}} + {x^{24}} + {x^{36}})$
$ \Rightarrow {x^{24}}{(^{12}}{C_0}{ + ^{12}}{C_6}{ + ^{12}}{C_{12}})$ (Here we are finding the coefficient of ${x^{24}}$.)
(Here, $^{12}{C_0} = \dfrac{{12!}}{{12! \times 0!}}$ = 1, we know that 0! is 1 and $^{12}{C_{12}} = \dfrac{{12!}}{{0! \times 12!}}$ = 1)
$ \Rightarrow {x^{24}}(1{ + ^{12}}{C_6} + 1)$
$ \Rightarrow {x^{24}}{(^{12}}{C_6} + 2)$
Thus, the coefficient of ${x^{24}}$ is $^{12}{C_6} + 2$.

Hence, Option B is the correct option.

Note:
Students must know the binomial expansion on ${(x + y)^n}$. They should also take care while using the formula of the combination when they are solving for $^{12}{C_0}$&$^{12}{C_1}$. While solving this question, students should pick all the coefficients of \[{x^{24}}\] carefully. If any of the coefficients is left then you will get an incorrect answer. You might find Binomial expansion lengthy and tedious to calculate. But a binomial expression that has large power can be easily calculated with the help of the Binomial Theorem. $^n{C_0}$, $^n{C_1}$, $^n{C_2}$…., $^n{C_n}$ are called binomial coefficients and can represented by ${C_0}$, ${C_1}$, ${C_2}$, …., ${C_n}$. The total number of terms in the expansion of ${\left( {x + y} \right)^n}$ are $(n + 1)$. These are a few important things about binomial expansion. You should keep all these things in your mind while solving these types of questions.