Binomial Theorem
Binomial summation identity
nta_pyq_2025_apr
Grade 12
Question:
If$\sum$$r+1$11 11 , then$\alpha$is equal to : 10$10 -1$11$\alpha$-11 ( )$\cdot$$Cr+1 = r=0$r 10 10 10
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for binomial summation identity and impose the required coefficient or exponent condition.
10 1r$r-1 - 1$11$\sum$( )$Cr+1$(4)$r=0$10 r 10 11 1 =$\sum$$(10 - )$$Cr+1$r 10$r=0$10$r+1$1$11 = 10$$\sum$$Cr+1 - 10$$\sum$($Cr+1$( ) ) 10$r=0$11 11$11 = 10$[$C_{1}$+$C_{2}$+$\ldots$. . +$C_{11}$] 1 2 11 1 1 1 11 11$11 - 10$[$C_{1}$( ) +$C_{2}$( ) +$\ldots$..+$C_{11}$( ) ] 10 10 10 11 1$11 = 10$[$2 - 1$] - 10 [$(1 + ) - 1$] 10 11 11$11 = 10(2) - 10$- + 10 10 10 11 11$(20) - 11 = 10$10 ∴$\alpha$= 20
Correct Answer: 4