There are 3 oranges, 5 apples and 6 mangoes in a fruit basket. Number of ways in which at least one fruit can be selected from the basket is

Options

A. 168
B. 167
C. 125
D. 124

Answer

To solve the problem of finding the number of ways to select at least one fruit from a basket containing 3 oranges, 5 apples, and 6 mangoes, we can follow these steps:

### Step 1: Determine the number of ways to select fruits
For each type of fruit, we can choose from 0 up to the maximum number available. The number of ways to select \( n \) identical items is given by \( n + 1 \) (including the option of selecting none).

- For **oranges**: There are 3 oranges, so the number of ways to select oranges is \( 3 + 1 = 4 \).
- For **apples**: There are 5 apples, so the number of ways to select apples is \( 5 + 1 = 6 \).
- For **mangoes**: There are 6 mangoes, so the number of ways to select mangoes is \( 6 + 1 = 7 \).

### Step 2: Calculate the total number of ways to select fruits
To find the total number of ways to select fruits (including the option of selecting none), we multiply the number of ways for each type of fruit:

\[ \text{Total ways} = (3 + 1)(5 + 1)(6 + 1) = 4 \times 6 \times 7 \]

### Step 3: Perform the multiplication
Calculating the product:

\[ 4 \times 6 = 24 \]
\[ 24 \times 7 = 168 \]

So, the total number of ways to select fruits, including the option of selecting none, is 168.

### Step 4: Exclude the case of selecting no fruits
Since we want the number of ways to select at least one fruit, we need to subtract the case where no fruit is selected (which is counted as 1 way):

\[ \text{Ways to select at least one fruit} = 168 - 1 = 167 \]

### Final Answer
The number of ways to select at least one fruit from the basket is **167**.

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