Question

There are $12$ fruits in a basket of which $5$ are apples, $4$ mangoes and $3$ bananas (fruits of same species are identical). How many ways are there to select at/east one fruit$?$

Answer 1

Let $x$ be the number of apples being selected
$y$ be the number of mangoes being selected and
$z$ be the number of bananas being selected.
Then $x=0,1,2,3,4,5$
$y=0,1,2,3,4$
$z=0,1,2,3$
Total number of triplets $(x,y,z)$ is $6\times 5\times 4=120$
Exclude $(0,0,0)$
$\therefore$ Number of combinations$=120-1=119.$