Home
Class 12
MATHS
If sum(r=0)^25 {.^50Cr*^(50-r)C(25-r)}=K...

If `sum_(r=0)^25 {.^50C_r*^(50-r)C_(25-r)}=K(.^50C_25),` then `K` is equal to

A

`2^(24)`

B

`2^(25)-1`

C

`(25)^(2)`

D

`2^(25)`

Text Solution

Verified by Experts

The correct Answer is:
D
`sum_(r=0)^(25)""^(50)C_(r).""^((50-r))C_(25-r)`
`sum_(r=0)^(25) (50!)/(r!(50-r)!)xx((50-r)!)/(25! xx (25-r)!)`
Now multiply and divide by 25!
`=sum_(r=0)^(25) (50!)/(r!) xx (25!)/(25!^(2).(25-r)!) = sum_(r=0)^(25) (50!)/(25! .25!)xx (25!)/(r!(25-r)!)=sum_(r=0)^(25)""^(50)C_(25).""^(25)C_(r)`
`""^(50)C_(25)sum_(r=0)^(25) ""^(25)C_(r)= ""^(50)C_(25).2^(25)=""^(50)C_(25).k`
Hence `k=2^(25)`
Promotional Banner

Similar Questions

Explore conceptually related problems

if sum_(r=0)^(25).^(50)C_(r)(.^(50-r)C_(25-r))=k(.^(50)C_25), then k equals: (a) 2^(25) (b) 2^(25)-1 (c) 2^(24) (d) 25^(2)

sum_(r=0)^(20).^(50-r)C_6

If sum_(r=1)^(10)r(r-1)10C_(r)=k.2^(9), then k is equal to- -1

The value of sum_(r=0)^(15)r^(2)((""^(15)C_(r))/(""^(15)C_(r-1))) is equal to

If (1-x^(2))^(n)=sum_(r=0)^(n)a_(r)x^(r)(1-x)^(2n-r), then a_(r) is equal to ^(n)C_(r) b.^(n)C_(r)3^(r) c.^(2n)C_(r) d.^(n)C_(r)2^(r)

" If "C_(r)=^(25)C_(r)" and "C_(0)+5.C_(1)+9.C_(2)+....+(101)*C_(25)=2^(25)k," then "k" is equal to "

The valur of sum_(r=0)^(n) sum_(p=0)^(r) ""^(n)C_(r) . ""^(r)C_(p) is equal to