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Binomial Theorem
Multinomial Expansion, Coefficient Extraction
jee_adv_2026_mock_p1
Grade 12

Question:

The coefficient of x^10 in the expansion of (1 + x + x^2 + x^3)^5 is:
A. 101
B. 102
C. 103
D. 104

Step-by-Step Solution

Key Concept: Use (1 + x + x^2 + x^3) = (1 - x^4)/(1 - x).
Step 1: (1 + x + x^2 + x^3)^5 = (1 - x^4)^5 (1 - x)^-5. Step 2: Coefficient of x^10 = Σ_{r=0}^{2} (-1)^r C(5,r) C(5 + 10 - 4r - 1, 10 - 4r). Step 3: = C(5,0)C(14,10) - C(5,1)C(10,6) + C(5,2)C(6,2) = 1001 - 1050 + 150 = 101.
Correct Answer: A
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