Binomial coefficients which are in decreasing order are

Options

A. `""^15C_5, ""^15C_6, ""^15C_7`
B. `""^15C_10, ""^15C_9, ""^15C_8`
C. `""^15C_6, ""^15C_7, ""^15C_8`
D. `""^15C_7, ""^15C_6, ""^15C_5`

Answer

To solve the problem of identifying the binomial coefficients that are in decreasing order, we will follow these steps:

### Step 1: Understand the Binomial Coefficient
The binomial coefficient \( \binom{n}{r} \) is calculated using the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items, and \( r \) is the number of items to choose.

### Step 2: Calculate Specific Binomial Coefficients
We will calculate the binomial coefficients for \( n = 15 \) and various values of \( r \) (from 5 to 10) to find their values.

1. **Calculate \( \binom{15}{5} \)**:
\[ \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15!}{5! \cdot 10!} \]
Simplifying:
\[ = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003 \]

2. **Calculate \( \binom{15}{6} \)**:
\[ \binom{15}{6} = \frac{15!}{6!(15-6)!} = \frac{15!}{6! \cdot 9!} \]
Simplifying:
\[ = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 5005 \]

3. **Calculate \( \binom{15}{7} \)**:
\[ \binom{15}{7} = \frac{15!}{7!(15-7)!} = \frac{15!}{7! \cdot 8!} \]
Simplifying:
\[ = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 6435 \]

4. **Calculate \( \binom{15}{8} \)**:
\[ \binom{15}{8} = \binom{15}{7} = 6435 \]

5. **Calculate \( \binom{15}{9} \)**:
\[ \binom{15}{9} = \binom{15}{6} = 5005 \]

6. **Calculate \( \binom{15}{10} \)**:
\[ \binom{15}{10} = \binom{15}{5} = 3003 \]

### Step 3: List the Values
Now, we have the following values:
- \( \binom{15}{5} = 3003 \)
- \( \binom{15}{6} = 5005 \)
- \( \binom{15}{7} = 6435 \)
- \( \binom{15}{8} = 6435 \)
- \( \binom{15}{9} = 5005 \)
- \( \binom{15}{10} = 3003 \)

### Step 4: Arrange in Decreasing Order
The values of the binomial coefficients in decreasing order are:
1. \( \binom{15}{7} = 6435 \)
2. \( \binom{15}{6} = 5005 \)
3. \( \binom{15}{5} = 3003 \)

### Step 5: Identify the Correct Option
Based on the calculated values, the correct order of binomial coefficients in decreasing order is:
- \( 6435, 5005, 3003 \)

Thus, the answer is:
- \( \binom{15}{7}, \binom{15}{6}, \binom{15}{5} \)