1.

If `f (n) = sum_(s=1)^n sum_(r=s)^n "^nC_r`` "^rC_s` , then `f(3) =`A. 27B. 19C. 1D. 5

Answer» Correct Answer - b
We have,
`f(n) = sum_(s=1)^(n) sum_(r=s)^(n) ""^(n)C_(r) ""^(r)C_(s)`
`rArr f(n) = sum_(r =1)^(n) ""^(n)C_(1) + sum_(r=2)^(n) ""^(n)C_(r) ""^(r)C_(2) +….+ sum_( r= n -1)^(n) ""^(n)C_(r) ""^(r)C_(n-1) + ""^(n)C_(n) ""^(n)C_(n) `
`rArr f(n) = ""^(n)C_(1)""^(1)C_(1)+ ""^(n)C_(2)(""^(2)C_(1) + ""^(2)C_(2)) + ""^(n)C_(3) (""^(3)C_(1) + ""^(n)C_(2))+""^(n)C_(3)(""^(3)C_(1)+""^(3)C_(2) + ""^(3)C_(n))`
` + ....+ ""^(n) C_(n) (""^(n)C_(1) + ""^(n)C_(2) +.... + ""^(n)C_(n))`
`rArr f(n) = ""^(n)C_(1)xx(2^(1) -1) + ""^(n)C_(2) xx(2^(2) -1) + ""^(n)C_(3) xx(2^(3) -1) +... + ""^(n)C_(n) xx(2^(n) -1)`
`rArr f(n) = sum_(r=1)^(n) = ""^(n)C_(r) 2^(r) - sum_(r=1)^(n) ""^(n)C_(r)`
`rArr f(n) = sum_(r=0)^(n) = ""^(n)C_(r) 2^(r) - sum_(r=1)^(n) ""^(n)C_(r)`
`rArr f(n) = 3^(n) -2^(n)`
`rArr f(3) = 3^(3) - 2^(3) = 19`


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