Question

An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at most three of them are red is

Answer 1

Number of ways to select at most $3$ red balls $=p\left(0\text{red ball}\right)+p\left(1\text{red ball}\right)+P\left(2\text{red balls}\right)+P\left(3\text{red balls}\right)$
$={}^7{C}_4+{}^5{C}_1\times {}^7{C}_3+{}^5{C}_2\times {}^7{C}_2+{}^5{C}_3\times {}^7{C}_1$
$=35+175+210+70=490$