The value of $^{50}C_4+\sum_{r=1}^{6}{}^{56-r}C_3$ is

JEE Main · 2005 · 08 May 2005 · Offline · Permutations and Combinations

The value of

^{50} C_{4} + \sum_{r = 1}^{6}^{56 - r} C_{3}

Answer: (D)

Writing out the sum, it is

(4 50) + (3 50) + (3 51) + (3 52) + (3 53) + (3 54) + (3 55)

. Use the identity

(3 n) + (4 n) = (4 n + 1)

repeatedly:

(4 50) + (3 50) = (4 51)

, then

(4 51) + (3 51) = (4 52)

, and continuing to absorb each successive term of the form

(3)

leads finally to

(4 55) + (3 55) = (4 56)

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option