The number of distinct terms in \[{\left( {a + b + c + d + e} \right)^3}\] is
A. 35
B. 38
C. 42
D. 45
Answer
611.1k+ views
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.
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.
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