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 \[{{2}^{12}}\].
\[\begin{align}
  & {{\Rightarrow }^{12}}{{C}_{0}}{{+}^{12}}{{C}_{1}}{{+}^{12}}{{C}_{2}}{{+}^{12}}{{C}_{3}}{{+}^{12}}{{C}_{4}}{{+}^{12}}{{C}_{5}}{{+}^{12}}{{C}_{6}}{{+}^{12}}{{C}_{7}}{{+}^{12}}{{C}_{8}}{{+}^{12}}{{C}_{9}}{{+}^{12}}{{C}_{10}}{{+}^{12}}{{C}_{11}}{{+}^{12}}{{C}_{12}}={{2}^{12}} \\
 & {{\Rightarrow }^{12}}{{C}_{1}}{{+}^{12}}{{C}_{2}}{{+}^{12}}{{C}_{3}}{{+}^{12}}{{C}_{4}}{{+}^{12}}{{C}_{5}}{{+}^{12}}{{C}_{6}}{{+}^{12}}{{C}_{7}}{{+}^{12}}{{C}_{8}}{{+}^{12}}{{C}_{9}}{{+}^{12}}{{C}_{10}}{{+}^{12}}{{C}_{11}}{{+}^{12}}{{C}_{12}}={{2}^{12}}-1 \\
\end{align}\]
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 \[={{2}^{12}}-1\].