If x, y and r are positive integralegers, then \[xC\_{r} +^... | MATH - DEFINITON OF COMBINATIONS, CONDITION COMBINATIONS, DIVISION INTO GROUPS, DERANGEMENTS | SolveeIt | Solveeit

Today, August 18

Question

If x, y and r are positive integers, then

$xC_{r} +^{x} ⥂ {C_{r - 1}}^{y}C_{1} +^{x} ⥂ {C_{r - 2}}^{y} ⥂ C_{2} + ..... +^{y} ⥂ C_{r} =$

A

$\frac{x!y!}{r!}$

B

$\frac{(x + y)!}{r!}$

C

$x + yC_{r}$

D

$xyC_{r}$

Answer

$x + yC_{r}$

Explanation

Solution

The result $x + yC_{r}$ is trivially true for $r = 1,2$ it can be easily proved by the principle of mathematical induction that the result is true for $r$ also.