JEE MainMathematics

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+4 marks−1 if incorrectNumericalPrevious-year question

Finding the Constant Term in a Multinomial Expansion

Q: Could there be other combinations of (r_1, r_2, r_3)?

No, because r_3 = 8r_2 - 5 with 0 ≤ r_3 ≤ 5 forces 5 ≤ 8r_2 ≤ 10, which leaves r_2 = 1 as the only possible integer.

The constant term in the expansion of (2 x+1/(x^7)+3 x^2)^5 is

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Step-by-step solutionView

Correct answer

The constant term in the expansion is 1080.

Option analysis

Why each option works or fails

Reviewed route

Solution

StepWorking

01Given

Expression: ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ .

02Goal

Find the constant term (the coefficient of $x^{0}$ ) in the expansion.

03Approach

Use the multinomial theorem. The general term is $\frac{5!}{r_{1}! r_{2}! r_{3}!} (2 x)^{r_{1}} (x^{- 7})^{r_{2}} (3 x^{2})^{r_{3}}$ , with non-negative integers satisfying $r_{1} + r_{2} + r_{3} = 5$ and power of $x$ equal to zero: $r_{1} - 7 r_{2} + 2 r_{3} = 0$ .

04Execute

From $r_{1} + r_{2} + r_{3} = 5$ and $r_{1} - 7 r_{2} + 2 r_{3} = 0$ , subtract the second from the first: $8 r_{2} - r_{3} = 5 ⟹ r_{3} = 8 r_{2} - 5$ . Since $r_{1}, r_{2}, r_{3} \in {0, 1, 2, 3, 4, 5}$ , if $r_{2} = 0$ , $r_{3} = - 5$ (invalid); if $r_{2} = 1$ , $r_{3} = 3$ , giving $r_{1} = 5 - 1 - 3 = 1$ ; if $r_{2} \geq 2$ , $r_{3} \geq 11 > 5$ (invalid). Thus, the unique solution is $(r_{1}, r_{2}, r_{3}) = (1, 1, 3)$ .

05Execute

Evaluate the coefficient for $(r_{1}, r_{2}, r_{3}) = (1, 1, 3)$ : $Term = \frac{5!}{1! \cdot 1! \cdot 3!} \cdot 2^{1} \cdot 1^{1} \cdot 3^{3} = \frac{120}{6} \cdot 2 \cdot 27 = 20 \cdot 54 = 1080$ .

✓Verify

Check the power of $x$ : $r_{1} - 7 r_{2} + 2 r_{3} = 1 - 7 (1) + 2 (3) = 1 - 7 + 6 = 0$ . Sum of powers: $1 + 1 + 3 = 5$ . The solution is unique and fully verified.

Hints that build this answer step by step

Using the multinomial theorem, what is the general term in the expansion of ${(2 x + x^{- 7} + 3 x^{2})}^{5}$ ?
$\frac{5!}{a! b! c!} 2^{a} 3^{c} x^{a - 7 b + 2 c}$ where $a + b + c = 5$
Which non-negative integer values of $(a, b, c)$ satisfy $a + b + c = 5$ and make the power of $x$ zero ( $a - 7 b + 2 c = 0$ )?
$(a, b, c) = (1, 1, 3)$ only
What is the value of the coefficient for $(a, b, c) = (1, 1, 3)$ ?
1080

✓ Source and academic review

Question type: Numerical
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 9 September 2026

Quick checks

Students also ask

Could there be other combinations of

(r_{1}, r_{2}, r_{3})

No, because $r_{3} = 8 r_{2} - 5$ with $0 \leq r_{3} \leq 5$ forces $5 \leq 8 r_{2} \leq 10$ , which leaves $r_{2} = 1$ as the only possible integer.

Answer

The constant term in the expansion is 1080.

Why each option works or fails

Step-by-step solution

Given: Expression: ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ .
Goal: Find the constant term (the coefficient of $x^{0}$ ) in the expansion.
Approach: Use the multinomial theorem. The general term is $\frac{5!}{r_{1}! r_{2}! r_{3}!} (2 x)^{r_{1}} (x^{- 7})^{r_{2}} (3 x^{2})^{r_{3}}$ , with non-negative integers satisfying $r_{1} + r_{2} + r_{3} = 5$ and power of $x$ equal to zero: $r_{1} - 7 r_{2} + 2 r_{3} = 0$ .
Execute: From $r_{1} + r_{2} + r_{3} = 5$ and $r_{1} - 7 r_{2} + 2 r_{3} = 0$ , subtract the second from the first: $8 r_{2} - r_{3} = 5 ⟹ r_{3} = 8 r_{2} - 5$ . Since $r_{1}, r_{2}, r_{3} \in {0, 1, 2, 3, 4, 5}$ , if $r_{2} = 0$ , $r_{3} = - 5$ (invalid); if $r_{2} = 1$ , $r_{3} = 3$ , giving $r_{1} = 5 - 1 - 3 = 1$ ; if $r_{2} \geq 2$ , $r_{3} \geq 11 > 5$ (invalid). Thus, the unique solution is $(r_{1}, r_{2}, r_{3}) = (1, 1, 3)$ .
Execute: Evaluate the coefficient for $(r_{1}, r_{2}, r_{3}) = (1, 1, 3)$ : $Term = \frac{5!}{1! \cdot 1! \cdot 3!} \cdot 2^{1} \cdot 1^{1} \cdot 3^{3} = \frac{120}{6} \cdot 2 \cdot 27 = 20 \cdot 54 = 1080$ .
Verify: Check the power of $x$ : $r_{1} - 7 r_{2} + 2 r_{3} = 1 - 7 (1) + 2 (3) = 1 - 7 + 6 = 0$ . Sum of powers: $1 + 1 + 3 = 5$ . The solution is unique and fully verified.

Finding the Constant Term in a Multinomial Expansion

What feels right?

The constant term in the expansion is 1080.

Why each option works or fails

Solution

01Given

02Goal

03Approach

04Execute

05Execute

✓Verify

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