Question

The number of combinations that can be formed with $5$ oranges, $4$ mangoes and $3$ bananas when it is essential to take
$\left(i\right)$ at least one fruit be "k"
$\left(ii\right)$ One fruit of each kind "m"
find $2m-k$ ?

Answer 1

Here, $5$ oranges are alike of one kind, $4$ mangoes are alike of second kind and $3$ bananas are alike of third kind.
$\left(i\right)$ the required number of combinations (at least one fruit)
$=(5+1)(4+1)(3+1)2^0-1=120-1=119=k$
$\left(ii\right)$ The required number of combinations (one fruit of each kind)
${=}^5{C}_1{\times }^4{C}_1{\times }^3{C}_1=5\times 4\times 3=60=m$

Then, $2m-k$
$=2\times 60-119=1$