Question

There are 4 mangoes, 3 apples, 2 oranges and 1 each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of each kind is
a. $10!$
b. $9!$
c. $4!$
d. none of these

Answer 1

Here in our given problem we have to choose at least 1 fruit each from a given number of fruits. Here we are going to use some general theorems of permutation and combination, and we are going to check which of our given options matches with the result we got.

Complete step-by-step answer:
We are to find the number of ways of selecting at least one fruit of each kind,
And we are given here 4 mangoes, 3 apples, 2 oranges and 1 each of other 3 varieties of fruits.
So, Now, the number of way of selecting 1 mango from 4 mangoes is, ${}^4{C_1}$,
And, again, the number of way of selecting 1 apple from 3 apples is ${}^3{C_1}$,
And, the number of selecting 1 orange from 2 oranges is ${}^2{C_1}$,
​And at the end, there is only one way selecting one fruit from one.
So, we will have,
Therefore, total no of ways of selection $= 4 \times 3 \times 2 \times 1 \times 1 \times 1 = 4!$
Thus, we have our answer as, $4!$which is option c.

Note: This is a general permutation and combination problem where we have used some general theorems of permutation and combination. We have used the case that if y objects are to be chosen from x objects , that can be done in, ${}^x{C_y}$number of ways.