In how many ways is it possible to make a selection by taking any number of 15 fruits, namely 3 oranges, 5 apples and 7 mangoes?

This problem can be solved using the concept of combinations in combinatorics.

The problem states that we can select any number of fruits from a total of 15 fruits, which includes 3 oranges, 5 apples, and 7 mangoes.

The key point to note here is that we can select 'any number' of fruits, which means we can also select 'none' of a particular type of fruit.

So, for each type of fruit, we have the following options:

• For oranges, we can select 0, 1, 2, or 3 oranges. That gives us 4 options.
• For apples, we can select 0, 1, 2, 3, 4, or 5 apples. That gives us 6 options.
• For mangoes, we can select 0, 1, 2, 3, 4, 5, 6, or 7 mangoes. That gives us 8 options.

Since these are independent choices, we multiply the number of options together to get the total number of ways to select any number of fruits.

So, the total number of ways = 4 options (for oranges) * 6 options (for apples) * 8 options (for mangoes) = 192 ways.

Therefore, there are 192 ways to make a selection by taking any number of the 15 fruits.