The power of 7 contained in `""^(1000)C_(500)` must be

Options

A. 3
B. 2
C. 1
D. 0

Answer

To find the power of 7 in the binomial coefficient \( \binom{1000}{500} \), we can use the formula for the power of a prime \( p \) in \( n! \):

\[ \text{Power of } p \text{ in } n! = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]

We will calculate the power of 7 in \( 1000! \), \( 500! \), and \( 500! \) separately, and then use the formula:

\[ \text{Power of } 7 \text{ in } \binom{n}{r} = \text{Power of } 7 \text{ in } n! - \text{Power of } 7 \text{ in } r! - \text{Power of } 7 \text{ in } (n-r)! \]

### Step 1: Calculate the power of 7 in \( 1000! \)

\[ \text{Power of } 7 \text{ in } 1000! = \left\lfloor \frac{1000}{7} \right\rfloor + \left\lfloor \frac{1000}{7^2} \right\rfloor + \left\lfloor \frac{1000}{7^3} \right\rfloor + \left\lfloor \frac{1000}{7^4} \right\rfloor \]

Calculating each term:

- \( \left\lfloor \frac{1000}{7} \right\rfloor = \left\lfloor 142.857 \right\rfloor = 142 \)
- \( \left\lfloor \frac{1000}{49} \right\rfloor = \left\lfloor 20.408 \right\rfloor = 20 \)
- \( \left\lfloor \frac{1000}{343} \right\rfloor = \left\lfloor 2.915 \right\rfloor = 2 \)
- \( \left\lfloor \frac{1000}{2401} \right\rfloor = 0 \) (as \( 2401 > 1000 \))

Adding these together:

\[ \text{Power of } 7 \text{ in } 1000! = 142 + 20 + 2 + 0 = 164 \]

### Step 2: Calculate the power of 7 in \( 500! \)

\[ \text{Power of } 7 \text{ in } 500! = \left\lfloor \frac{500}{7} \right\rfloor + \left\lfloor \frac{500}{7^2} \right\rfloor + \left\lfloor \frac{500}{7^3} \right\rfloor + \left\lfloor \frac{500}{7^4} \right\rfloor \]

Calculating each term:

- \( \left\lfloor \frac{500}{7} \right\rfloor = \left\lfloor 71.428 \right\rfloor = 71 \)
- \( \left\lfloor \frac{500}{49} \right\rfloor = \left\lfloor 10.204 \right\rfloor = 10 \)
- \( \left\lfloor \frac{500}{343} \right\rfloor = \left\lfloor 1.459 \right\rfloor = 1 \)
- \( \left\lfloor \frac{500}{2401} \right\rfloor = 0 \)

Adding these together:

\[ \text{Power of } 7 \text{ in } 500! = 71 + 10 + 1 + 0 = 82 \]

### Step 3: Calculate the power of 7 in \( \binom{1000}{500} \)

Now we can find the power of 7 in \( \binom{1000}{500} \):

\[ \text{Power of } 7 \text{ in } \binom{1000}{500} = \text{Power of } 7 \text{ in } 1000! - 2 \times \text{Power of } 7 \text{ in } 500! \]

Substituting the values we calculated:

\[ \text{Power of } 7 \text{ in } \binom{1000}{500} = 164 - 2 \times 82 = 164 - 164 = 0 \]

### Final Answer

The power of 7 contained in \( \binom{1000}{500} \) is \( 0 \).

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