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

Hint: 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.