Question

The number of distinct terms in ${\left( {a + b + c + d + e} \right)^3}$ is
A. 35
B. 38
C. 42
D. 45

Answer 1

As we have to find the number of terms so, we will use the permutation and combination concept. We have number of variables in the bracket equal to 5 and value of $r$ is given as 3 so, we will use the formula, ${}^{n + r - 1}{C_{n - 1}}$ to find the numbers of distinct terms of non-integral solution. We will use the formula $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ to solve the expression ${}^n{C_r}$. Hence, we will get the desired result.

Complete step by step answer:

We will first consider the given expression ${\left( {a + b + c + d + e} \right)^3}$ and we have to find the number of distinct terms.
Since the number of terms of non-integral solution $a + b + c + d + e$ is given by, ${}^{n + r - 1}{C_{n - 1}}$.
Therefore, the value of $n$ is 5, and the value of $r$ is given by 3 that is the power on the given expression.
Thus, we will substitute the values in the expression and thus we get,
$\Rightarrow {}^{5 + 3 - 1}{C_{5 - 1}} = {}^7{C_4}$
Now, we will use the formula $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ for the expansion of obtained value.
Thus, we get,
$\Rightarrow \dfrac{{7!}}{{4!3!}} = \dfrac{{7 \times 6 \times 5}}{{3 \times 2}} = 35$
Hence, we can conclude that the number of distinct terms in the given expression is 35.
Thus, option A is correct.

Note: Remember the expansion of ${}^n{C_r}$ which is given by $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$. For obtaining the number of terms for a non-integral solution, we have to use ${}^{n + r - 1}{C_{n - 1}}$. Do not make mistakes in solving the factorial value. For such questions, remember the concepts of permutation and combination. Substitute the values in the formula properly while doing the calculations.