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# A bag contains 3 black 4 white and 2 red balls, all balls being different. The number of selection of at the most 6 balls so that each selection has balls of all the 3 colours is ?

Since each selection should have all the three colours, minimum 3 balls one
in each colour should be selected in $^3C_1\times ^4C_1\times ^2C_1$ ways.
Maximum 6 balls are to be selected.
$\therefore$ The required no. of ways =$^3C_1\times ^4C_1\times ^2C_1(^6C_0+^6C_1+^6C_2+^6C_3)$ ways
$=24(1+6+15+20)=42.4!$