Question

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

Answer 1

Hint: We know that the number of ways to pick r objects from n objects is equal to $^{n}{{C}_{r}}$. So, we have to find the number of ways to pick $i$ fruits from 12 fruits where $(1\le i\le 12)$. So, we have to find the sum of the number of ways to pick $i$ fruits from 12 fruits where $(1\le i\le 12)$. This gives us the number of ways to select at least one fruit from 12 fruits in a basket.

Complete step by step solution:
From the question, we are given that there are 12 fruits in a basket. In this basket, we are having 5 apples, 4 mangoes and 3 bananas. We are also given that fruits of the same species are different. We have to find the number of ways to pick at least one fruit from 12 fruits in a basket.
We know that the number of ways to pick r objects from n objects is equal to $^{n}{{C}_{r}}$.
We have to pick at least one fruit from the basket. It means that we can pick r fruits from the basket $(1\le r\le n)$.
So, the number of ways to pick 1 fruit from 12 fruits ${{=}^{12}}{{C}_{1}}$.
In the similar way, the number of ways to pick 2 fruits from 12 fruits ${{=}^{12}}{{C}_{2}}$.
In the similar way, the number of ways to pick 3 fruits from 12 fruits ${{=}^{12}}{{C}_{3}}$.
In the similar way, the number of ways to pick 4 fruits from 12 fruits ${{=}^{12}}{{C}_{4}}$.
In the similar way, the number of ways to pick 5 fruits from 12 fruits ${{=}^{12}}{{C}_{5}}$.
In the similar way, the number of ways to pick 6 fruits from 12 fruits ${{=}^{12}}{{C}_{6}}$.
In the similar way, the number of ways to pick 7 fruits from 12 fruits ${{=}^{12}}{{C}_{7}}$.
In the similar way, the number of ways to pick 8 fruits from 12 fruits ${{=}^{12}}{{C}_{8}}$.
In the similar way, the number of ways to pick 9 fruits from 12 fruits ${{=}^{12}}{{C}_{9}}$.
In the similar way, the number of ways to pick 10 fruits from 12 fruits ${{=}^{12}}{{C}_{10}}$.
In the similar way, the number of ways to pick 11 fruits from 12 fruits ${{=}^{12}}{{C}_{11}}$.
In the similar way, the number of ways to pick 12 fruits from 12 fruits ${{=}^{12}}{{C}_{12}}$
The total number of ways to pick at least one fruit
${{=}^{12}}{{C}_{1}}{{+}^{12}}{{C}_{2}}{{+}^{12}}{{C}_{3}}{{+}^{12}}{{C}_{4}}{{+}^{12}}{{C}_{5}}{{+}^{12}}{{C}_{6}}{{+}^{12}}{{C}_{7}}{{+}^{12}}{{C}_{8}}{{+}^{12}}{{C}_{9}}{{+}^{12}}{{C}_{10}}{{+}^{12}}{{C}_{11}}{{+}^{12}}{{C}_{12}}$ $=\sum\limits_{i=1}^{12}{^{12}{{C}_{r}}}$
We know that the sum of coefficients of ${{(1+x)}^{n}}$ is equal to ${{2}^{n}}$.
In the same way, the sum of coefficients of ${{(1+x)}^{12}}$ is equal to ${{2}^{12}}$.

& {{\Rightarrow }^{12}}{{C}_{0}}{{+}^{12}}{{C}_{1}}{{+}^{12}}{{C}_{2}}{{+}^{12}}{{C}_{3}}{{+}^{12}}{{C}_{4}}{{+}^{12}}{{C}_{5}}{{+}^{12}}{{C}_{6}}{{+}^{12}}{{C}_{7}}{{+}^{12}}{{C}_{8}}{{+}^{12}}{{C}_{9}}{{+}^{12}}{{C}_{10}}{{+}^{12}}{{C}_{11}}{{+}^{12}}{{C}_{12}}={{2}^{12}} \\\ & {{\Rightarrow }^{12}}{{C}_{1}}{{+}^{12}}{{C}_{2}}{{+}^{12}}{{C}_{3}}{{+}^{12}}{{C}_{4}}{{+}^{12}}{{C}_{5}}{{+}^{12}}{{C}_{6}}{{+}^{12}}{{C}_{7}}{{+}^{12}}{{C}_{8}}{{+}^{12}}{{C}_{9}}{{+}^{12}}{{C}_{10}}{{+}^{12}}{{C}_{11}}{{+}^{12}}{{C}_{12}}={{2}^{12}}-1 \\\ \end{aligned}$$ So, the total number of ways to pick at least one fruit $$={{2}^{12}}-1$$. Note: This problem can be solved in an alternative method also. Total number of ways to pick at least one fruit from 12 fruits in the basket = Total number of ways to pick any number of fruits in the basket – total number of ways to pick 0 fruits from the basket. We know that the number of ways to pick r objects from n objects is equal to $$^{n}{{C}_{r}}$$. In the similar manner, the number of ways to pick r objects from n objects is equal to $$^{12}{{C}_{0}}$$. We know that the total number of ways to pick any number of objects from n objects is equal to $${{2}^{n}}$$. In the similar manner, the number of ways to pick any number of fruits from 12 fruits is equal to $${{2}^{12}}$$. Total number of ways to pick at least one fruit from 12 fruits in the basket $$={{2}^{12}}{{-}^{12}}{{C}_{0}}$$. Total number of ways to pick at least one fruit from 12 fruits from the basket $$={{2}^{12}}-1$$.