If 
then the value of k is :
Text Solution
Verified by ExpertsC
Numerator = n C r + n C r+1 + 3( n C r+1 + n C r+2 ) + 3( n C r+2 + n C r+3 ) + n C r+3 + n C r+4
= n+1 C r+1 + 3 n+1 C r+2 + 3 n+1 C r+3 + n+1 C r+4 = n+1 C r+1 + n+1 C r+2 + 2( n+1 C r+2 + n+1 C r+3 ) + n+1 C r+3 + n+1 C r+4
= n+2 C r+2 + 2 n+2 C r+3 + n+2 C r+4 = n+2 C r+2 + n+2 C r+3 + n+2 C r+4 + n+2 C r+3 = n+3 C r+3 + n+3 C r+4 = n+4 C r+4
Denominator = n C r + n C r+1 + 2 ( n C r+1 + n C r+2 ) + ( n C r+2 + n C r+3 ) = n+1 C r+1 + 2 n+1 C r+2 + n+1 C r+3
= n+1 C r+1 + n+1 C r+2 + n+1 C r+2 + n+1 C r+3 = n+2 C r+2 + n+2 C r+3 = n+3 C r+3
∴ The expression is equal to : 
= 
∴ k = 4
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