Answer these questions based on the following information.
A shop has 10 mobile recharge coupons of Vodafone. Out of which 4 are identical of same value and remaining each coupons is of distinct value.
Find the number of ways of selecting 3 coupons.

Options

A. 6
B. 21
C. 22
D. 42

Answer

To solve the problem of selecting 3 coupons from a total of 10 coupons where 4 are identical and the remaining 6 are distinct, we can break it down into different cases based on how many identical coupons we select.

### Step-by-Step Solution:

1. **Identify the Types of Selections**:
- We can select 3 coupons in different ways based on the number of identical coupons chosen:
- Case 1: All 3 coupons are identical.
- Case 2: 2 coupons are identical, and 1 coupon is distinct.
- Case 3: 1 coupon is identical, and 2 coupons are distinct.
- Case 4: All 3 coupons are distinct.

2. **Calculate Each Case**:
- **Case 1**: Selecting 3 identical coupons.
- Since all 3 are identical, there is only **1 way** to select them.
- Total for Case 1 = 1.

- **Case 2**: Selecting 2 identical coupons and 1 distinct coupon.
- We can choose 1 distinct coupon from the 6 available.
- The number of ways to choose 1 from 6 is given by \( \binom{6}{1} = 6 \).
- Total for Case 2 = 6.

- **Case 3**: Selecting 1 identical coupon and 2 distinct coupons.
- We can choose 2 distinct coupons from the 6 available.
- The number of ways to choose 2 from 6 is given by \( \binom{6}{2} = 15 \).
- Total for Case 3 = 15.

- **Case 4**: Selecting 3 distinct coupons.
- We can choose 3 distinct coupons from the 6 available.
- The number of ways to choose 3 from 6 is given by \( \binom{6}{3} = 20 \).
- Total for Case 4 = 20.

3. **Sum Up All Cases**:
- Total number of ways to select 3 coupons = Case 1 + Case 2 + Case 3 + Case 4
- Total = \( 1 + 6 + 15 + 20 = 42 \).

### Final Answer:
The total number of ways to select 3 coupons is **42**.

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