If$\sum$9 then ($\alpha$+$\beta$) is equal to 9$r+3$9 3 2 ( r ). Cr =$\alpha$( ) -$\beta$,$\alpha$,$\beta$$\ in $N,$r=1$2 2
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for binomial coefficient sum and impose the required coefficient or exponent condition.
$\sum$9 ($r+3$)$\cdot$9 Cr =$\alpha$( 3 ) 9 -$\beta$,$\alpha$,$\beta$$\ in $N$r=1$r 2 2 (3) Now, 9 9 9$r + 3$r 3 9 9 9$\sum$( )$\cdot$Cr =$\sum$( )$\cdot$Cr +$\sum$( )$\cdot$Cr r r r 2 2 2$r=1$$r=1$$r=1$9 9 r 9 9 1 Cr 9 8 9 =$\sum$( )$\cdot$$Cr-1 + 3$$\sum$Cr ( ) [U sin g = ] r 2 2 8$Cr-1$r$r=1$$r=1$9$r-1$9 r 9 1 1 8 9 =$\sum$$Cr-1$($) + 3$($\sum$( C r ( )$) - 1)$2 2 2$r=1$$r=0$8 9 9 1$1 = (1 + ) + 3$$((1 + ) - 1)$2 2 2 8 9 9 9 3 3 3 =$\cdot$($) + 3($$) - 3 = 6$$\cdot$($) - 3$2 2 2 2 Hence,$\alpha$= 6,$\beta$= 3 Thus ($\alpha$+$\beta$) = 81 2
Correct Answer: 3