Question

In how many ways can we put five identical fruits into three bowls? Note that the bowls may be empty.

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Answer 1

Explanation:

Let F={A,B,C,D,E} be the fruits.

Consider the following string

ABCDE||

permutating this you'll get the number that you're looking for: 7!2!

The string "ABCDE||" means all the fruits in the first bowl. And "A|B|CDE" means A in the first, B in the second and C,D,E in the latter

Answer 2

The number of ways that we can put five identical fruits into three bowls is: 21 ways

How to Solve Permutation?

There are different ways to approach this problem, but one possible method is to use stars and bars.

We can represent the five fruits as five stars and the three bowls as two dividers (bars), which create three sections to place the stars.

For example, if we have two stars in the first section, one star in the second section, and two stars in the third section, this corresponds to putting two fruits in the first bowl, one fruit in the second bowl, and two fruits in the third bowl.

The number of ways to arrange five stars and two bars is the same as the number of ways to choose two positions out of seven to place the bars, which is given by the binomial coefficient;ways

(5 + 3 - 1) C (3 - 1) = 7C2 = 21