JEE Main 2021 · 25 July · morning · Numerical answer · +4 / −0
The ratio of the coefficient of the middle term in the expansion of (1 + x)²⁰ and the sum of the coefficients of two middle terms in expansion of (1 + x)¹⁹ is _____________.
Answer and solution
Answer 1
Solution
Maths sir · 4 steps
No big numbers needed: by Pascal's rule, the two middle coefficients of 19 add up to the middle one of 20.
- Middle term for 20. (1+x)²⁰ has 21 terms, an odd count, so there is one middle term, T₁₁. Its coefficient is C(20, 10).
- Middle terms for 19. (1+x)¹⁹ has 20 terms, an even count, so there are two middle terms, T₁₀ and T₁₁. Their coefficients are C(19, 9) and C(19, 10).
- Pascal's rule. Two neighbouring coefficients in row n add up to the coefficient just below them: C(n, r-1) + C(n, r) = C(n+1, r). Take n = 19 and r = 10.
C(20, 10)/(C(19, 9)+C(19, 10)) = C(20, 10)/C(20, 10) = 1
- So. The two quantities are equal, so the ratio is 1.
The common slip: 2. Using only one middle term of (1+x)¹⁹ gives C(20, 10)/C(19, 9) = 20/10 = 2. The question adds both middle terms.
The idea to keep. For (1+x)ⁿ: if n is even there is one middle term, T_(n/2+1). If n is odd there are two. When two adjacent C(n, r) are added, use Pascal's rule before any arithmetic.
Answer from the NTA final answer key. Solution by PrepWiser.