Home
Class 12
MATHS
Evaluate ^(47)C(4)+sum(j=0)^(3)""^(50-...

Evaluate
`^(47)C_(4)+sum_(j=0)^(3)""^(50-j)C_(3)+sum_(k=0)^(5) ""^(56-k)C_(53-k)`.

Text Solution

Verified by Experts

We have, `.^(47)C_(4)+underset(j=0)overset(3)(sum).^(50-j)C_(3)+underset(k=0)overset(5)(sum).^(56-k)C_(53-k)`
`=.^(47)C_(4)+underset(j=0)overset(3)(sum).^(50-j)C_(3)+underset(k=0)overset(5)(.^(56)-k)C_(3)" "[because.^(n)C_(r)=.^(n)C_(n-r)]`
`=.^(47)C_(4)+(.^(50)C_(3)+.^(49)C_(3)+.^(48)C_(3)+.^(47)C_(3))`
`+(.^(56)C_(3)+.^(55)C_(3)+.^(54)C_(3)+.^(53)C_(3)+.^(52)C_(3)+.^(51)C_(3))`
`=.^(47)C_(4)+.^(47)C_(3)+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+.^(52)C_(3)+.^(52)C_(3)+.^(54)C_(3)+.^(55)C_(3)+.^(56)C_(3)`
`=(.^(47)C_(4)+.^(47)C_(3))+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+.^(52)C_(3)+.^(53)C_(3)+.^(54)C_(3)+.^(55)C_(3)+.^(56)C_(3)`
`=.^(48)C_(4)+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+ . . .+.^(56)C_(3)`
`=.^(49)C_(4)+.^(49)C_(3)+.^(50)C_(3)+ . . .+.^(56)C_(3)`
`=.^(56)C_(4)+.^(56)C_(3)=.^(57)C_(4)`
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the value of sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo)(1)/(3^(i)3^(j)3^(k)) .

sum_(r=0)^(n).^(n)C_(r)4^(r)=..........

Evaluate sum_(k-1)^11(2+3^k)

The sum of the series sum_(r=0)^(10) .^(20)C_(r) , is 2^(19)+{(.^(20)C_(10))/2} .

The value of lim_(n->oo)sum_(k=1)^n(6^k)/((3^k-2^k)(3^(k+1)-2^(k+1)) is equal to

Let S_(n)=sum_(k=1)^(4n)(-1)^((k(k+1))/(2))*k^(2) , then S_(n) can take value

Let S_k ,k=1,2, ,100 , denotes thesum of the infinite geometric series whose first term s (k-1)/(k !) and the common ratio is 1/k , then the value of (100^2)/(100 !)+sum_(k=2)^(100)(k^2-3k+1)S_k is _______.

Definite integration as the limit of a sum : lim_(ntooo)sum_(k=0)^(n)(n)/(n^(2)+k^(2))=..........

Statement 1 The sum of the series 1+(1+2+4)+(4+6+9)+(9+12+16)+"……."+(361+380+400) is 8000. Statement 2 sum_(k=1)^(n)(k^(3)-(k-1)^(3))=n^(3) for any natural number n.