Question

In the expansion of ${\left( {1 + x} \right)^n}.{\left( {1 + y} \right)^n}.{\left( {1 + z} \right)^n}$ the sum of the coefficients of the terms of degree r is
$\left( A \right){\left( {{}^n{C_r}} \right)^3}$
$\left( B \right)3\left( {{}^n{C_r}} \right)$
$\left( C \right)\left( {{}^{3n}{C_r}} \right)$
$\left( D \right)\left( {{}^n{C_{3r}}} \right)$

Answer 1

Hint – In this particular question use the concept of Binomial theorem i.e. the expansion of ${\left( {1 + a} \right)^n}$ is given as
${\left( {1 + a} \right)^n} = {}^n{C_o} + {}^n{C_1}\left( a \right) + {}^n{C_2}{\left( a \right)^2} + {}^n{C_3}{\left( a \right)^3} + .......... + {}^n{C_r}{\left( a \right)^r} + ....... + {}^n{C_n}{\left( a \right)^n}$ so when we multiply three terms like this so the number of terms in the expansion is 3n terms, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
According to Binomial theorem the expansion of ${\left( {1 + a} \right)^n}$ is given as,
$ \Rightarrow {\left( {1 + a} \right)^n} = {}^n{C_o} + {}^n{C_1}\left( a \right) + {}^n{C_2}{\left( a \right)^2} + {}^n{C_3}{\left( a \right)^3} + .......... + {}^n{C_r}{\left( a \right)^r} + ....... + {}^n{C_n}{\left( a \right)^n}$
Where, r is the ${r^{th}}$ term in the expansion and n is the ${n^{th}}$ term in the expansion.
So according to this Binomial theorem expand the given equation we have,
${\left( {1 + x} \right)^n}.{\left( {1 + y} \right)^n}.{\left( {1 + z} \right)^n}$
$ \Rightarrow {\left( {1 + x} \right)^n} = {}^n{C_o} + {}^n{C_1}\left( x \right) + {}^n{C_2}{\left( x \right)^2} + {}^n{C_3}{\left( x \right)^3} + .......... + {}^n{C_r}{\left( x \right)^r} + ....... + {}^n{C_n}{\left( x \right)^n}$.... (1)
$ \Rightarrow {\left( {1 + y} \right)^n} = {}^n{C_o} + {}^n{C_1}\left( y \right) + {}^n{C_2}{\left( y \right)^2} + {}^n{C_3}{\left( y \right)^3} + .......... + {}^n{C_r}{\left( y \right)^r} + ....... + {}^n{C_n}{\left( y \right)^n}$.... (2)
$ \Rightarrow {\left( {1 + z} \right)^n} = {}^n{C_o} + {}^n{C_1}\left( z \right) + {}^n{C_2}{\left( z \right)^2} + {}^n{C_3}{\left( z \right)^3} + .......... + {}^n{C_r}{\left( z \right)^r} + ....... + {}^n{C_n}{\left( z \right)^n}$.... (3)
Where, r is the ${r^{th}}$ term in the expansion and n is the ${n^{th}}$ term in the expansion.
Now multiply these equations we have,
$ \Rightarrow {\left( {1 + x} \right)^n}.{\left( {1 + y} \right)^n}.{\left( {1 + z} \right)^n} = \left[ {{}^n{C_o} + {}^n{C_1}\left( x \right) + {}^n{C_2}{{\left( x \right)}^2} + ... + {}^n{C_r}{{\left( x \right)}^r} + ... + {}^n{C_n}{{\left( x \right)}^n}} \right]$
                                                                         . \[\left[ {{}^n{C_o} + {}^n{C_1}\left( y \right) + {}^n{C_2}{{\left( y \right)}^2} + ... + {}^n{C_r}{{\left( y \right)}^r} + ... + {}^n{C_n}{{\left( y \right)}^n}} \right]\]
                                                                          . $\left[ {{}^n{C_o} + {}^n{C_1}\left( z \right) + {}^n{C_2}{{\left( z \right)}^2} + ... + {}^n{C_r}{{\left( z \right)}^r} + ... + {}^n{C_n}{{\left( z \right)}^n}} \right]$
So as we see that in the expansion of ${\left( {1 + x} \right)^n}.{\left( {1 + y} \right)^n}.{\left( {1 + z} \right)^n}$ each term has n terms so the total number of terms of degree r when multiplied together are 3n terms.
So the sum of the coefficient of the terms of degree r is given as
${}^{3n}{C_r}$
So this is the required answer.
Hence option (C) is the correct answer.

Note – Whenever we face such types of question the key concept we have to remember is that the expansion of ${\left( {1 + a} \right)^n}$ according to Binomial theorem which is all stated above so first write the expansion s above then multiply it together as above then the number of terms in the expansion of the given equation is 3n terms as every expansion has n terms in the expansion for example (a +b) when multiply by (c + d) gives 4 terms and every equation has 2 terms so the sum of the coefficient of the terms of degree r is ${}^{3n}{C_r}$.