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 7 coupons.

Options

A. 3
B. 21
C. 42
D. 22

Answer

To solve the problem of selecting 7 coupons from a total of 10 coupons (where 4 are identical and the remaining 6 are distinct), we can break it down into cases based on how many identical coupons we select.

### Step-by-Step Solution:

1. **Identify the Total Coupons**: We have 10 coupons in total:
- 4 identical coupons (let's call them A)
- 6 distinct coupons (let's call them B1, B2, B3, B4, B5, B6)

2. **Define Cases Based on Identical Coupons**: We will consider different cases based on how many identical coupons we select. The possible cases are:
- Case 1: Select 4 identical coupons (A) and 3 distinct coupons from the 6 available.
- Case 2: Select 3 identical coupons (A) and 4 distinct coupons from the 6 available.
- Case 3: Select 2 identical coupons (A) and 5 distinct coupons from the 6 available.
- Case 4: Select 1 identical coupon (A) and 6 distinct coupons from the 6 available.
- Case 5: Select 0 identical coupons (not possible since we need to select 7 coupons).

3. **Calculate Each Case**:
- **Case 1**: 4 identical + 3 distinct
- Number of ways to choose 3 distinct from 6: \( \binom{6}{3} \)
- **Case 2**: 3 identical + 4 distinct
- Number of ways to choose 4 distinct from 6: \( \binom{6}{4} \)
- **Case 3**: 2 identical + 5 distinct
- Number of ways to choose 5 distinct from 6: \( \binom{6}{5} \)
- **Case 4**: 1 identical + 6 distinct
- Number of ways to choose 6 distinct from 6: \( \binom{6}{6} \)

4. **Calculate the Binomial Coefficients**:
- \( \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \)
- \( \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \)
- \( \binom{6}{5} = \frac{6!}{5!(6-5)!} = 6 \)
- \( \binom{6}{6} = 1 \)

5. **Sum the Cases**:
- Total ways = \( \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6} \)
- Total ways = \( 20 + 15 + 6 + 1 = 42 \)

### Final Answer:
The total number of ways to select 7 coupons is **42**.

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