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JEE Main202621 January 2026Evening ShiftMathematicsBinomial TheoremActual

If ( 1 ¹⁵ C ₀ + 1 ¹⁵ C ₁ ) ( 1 ¹⁵ C ₁ + 1 ¹⁵ C ₂ ) ( 1 ¹⁵ C ₁₂ + 1 ¹⁵ C ₁₃ )= ¹³ ¹⁴ C ₀ ¹⁴ C ₁ ¹⁴ C ₁₂ , then 30 is equal to _ _ _ _ .

Correct answer

0

Step-by-step solution

The general term of the product is T_r = 1 ¹⁵C_ r-1 + 1 ¹⁵C_r for r = 1, 2, , 13 . Using the formula ^ n C_r = n r ^ n-1 C_ r-1 , we have ¹⁵C_r = 15 r ¹⁴C_ r-1 and ¹⁵C_ r-1 = 15 15-(r-1) ¹⁴C_ r-1 = 15 16-r ¹⁴C_ r-1 . Substituting these into T_r : T_r = 16-r 15 ¹⁴C_ r-1 + r 15 ¹⁴C_ r-1 = 16-r+r 15 ¹⁴C_ r-1 = 16 15 ¹⁴C_ r-1 . The product is _ r=1 ¹³ T_r= _ r=1 ¹³ 16 15 ¹⁴C_ r-1 = 16¹³ 15¹³ _ r=1 ¹³ ¹⁴C_ r-1 . The denominator of the product is 15¹³ (¹⁴C₀ ¹⁴C₁ ¹⁴C₁₂) . Comparing this with the given expression ¹³ ¹⁴C₀ ¹

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