A gym opens twice a day - morning and evening. But I don't work out twice a day. I attend it only 4 times a week. In how many ways can I attend the gym in a week?

Options

A. 560
B. 350
C. 280
D. none of these

Answer

To solve the problem of how many ways you can attend the gym in a week, we can break it down into a series of steps:

### Step-by-Step Solution:

1. **Identify the Requirements**:
- You attend the gym only 4 times a week.
- You can attend either in the morning or the evening, but not both on the same day.

2. **Determine the Total Days**:
- There are 7 days in a week.

3. **Select Days to Attend**:
- You need to choose 4 days out of the 7 days to attend the gym.
- This can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of days (7) and \( r \) is the number of days you will attend (4).
- The formula is given by:
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} \]

4. **Calculate the Combinations**:
- Calculate \( \binom{7}{4} \):
\[ \binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

5. **Determine Time Slots**:
- For each of the 4 selected days, you have 2 options (morning or evening).
- Therefore, the total number of ways to choose the time slots for the 4 days is:
\[ 2^4 = 16 \]

6. **Calculate Total Combinations**:
- Now, multiply the number of ways to choose the days by the number of ways to choose the time slots:
\[ \text{Total Ways} = \binom{7}{4} \times 2^4 = 35 \times 16 = 560 \]

### Final Answer:
The total number of ways you can attend the gym in a week is **560**.

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