Series of Binomial Coefficients

JEE Main Mathematics · Binomial Theorem · 15 questions, latest first

• If 26(2^3/3(^12 C_2)+2^5/5(^12 C_4)+2^7/7(^12 C_6)+⋯+(2^13)/13(^12 C_12))=3^13-α, then α is equal to:20268 April, Shift 2 · Q8
• The value of (^100 C_50)/51+(^100 C_51)/52+….+(^100 C_100)/101 is:202623 January, Shift 1 · Q7
• The sum of the series 2 × 1 ×^20 C_4-3 × 2 ×^20 C_5+4 × 3 ×^20 C_6-5 × 4 ×^20 C_7+⋯ ⋯+18 × 17 ×^20 C_20, is equal to ____.20257 April, Shift 2 · Q21
• If 1^2 ·(^15 C_1)+2^2 ·(^15 C_2)+3^2 ·(^15 C_3)+…+15^2 ·(^15 C_15)=2^m · 3^n · 5^k, where m, n, k ∈ N, then m+n+k is equal to:20254 April, Shift 2 · Q7
• If Σ_r=1^9((r+3)/2^r) ·^9 C_r=α(3/2)^9-β, α, β ∈ N, then (α+β)^2 is equal to20253 April, Shift 1 · Q8
• If Σ_r=0^10((10^r+1-1)/(10^r)) ·^11 C_r+1=(α^11-11^11)/(10^10), then α is equal to:20252 April, Shift 2 · Q7
• If Σ_r=1^30 (r^2(^30 C_r)^2)/(^30 C_r-1)=α × 2^29, then α is equal to ____.202522 January, Shift 2 · Q21
• If Σ_r=0^5 (^11 C_2 r+1)/(2 r+2)=m/n, gcd(m, n)=1, then m-n is equal to ____.202522 January, Shift 1 · Q22
• Let α=Σ_r=0^n(4 r^2+2 r+1)^n C_r and β=(Σ_r=0^n (^n C_r)/(r+1))+1/(n+1). If 140<(2 α)/(β)<281, then the value of n is ____.20248 April, Shift 1 · Q24
• Let α=Σ_k=0^n(((^n C_k)^2)/(k+1)) and β=Σ_k=0^n-1((^n C_k^n C_k+1)/(k+2)). If 5 α=6 β, then n equals ____.202430 January, Shift 2 · Q24
• If (^11 C_1)/2+(^11 C_2)/3+…+(^11 C_9)/10=n/m with gcd(n, m)=1, then n+m is equal to ____.202429 January, Shift 1 · Q23
• If 1/(n+1)^n C_n+1/n^n C_n-1+…+1/2^n C_1+^n C_0=1023/10 then n is equal to202312 April, Shift 1 · Q6
• If (^30 C_1)^2+2(^30 C_2)^2+3(^30 C_3)^2+…+30(^30 C_30)^2=(α 60!)/((30!)^2) then α is equal to:202324 January, Shift 2 · Q68
• The value of Σ_r=0^22^22 C_r^23 C_r is202324 January, Shift 1 · Q69
• Suppose Σ_r=0^2023 r^2^2023 C_r=2023 × α × 2^2022. Then the value of α is ____202324 January, Shift 1 · Q84