The exponent of 7 in `100C_50` is

Explanation

To find the exponent of 7 in \( \binom{100}{50} \), we can use the formula for combinations:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In this case, we have:

\[ \binom{100}{50} = \frac{100!}{50! \times 50!} \]

To find the exponent of 7 in \( \binom{100}{50} \), we need to calculate the exponent of 7 in \( 100! \) and subtract the exponent of 7 in \( 50! \) (twice, since it appears in both \( 50! \) terms).

### Step 1: Calculate the exponent of 7 in \( 100! \)

Using the formula for the maximum power of a prime \( p \) in \( n! \):

\[ \text{Exponent of } p \text{ in } n! = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots \]

For \( n = 100 \) and \( p = 7 \):

\[ \text{Exponent of } 7 \text{ in } 100! = \left\lfloor \frac{100}{7} \right\rfloor + \left\lfloor \frac{100}{7^2} \right\rfloor + \left\lfloor \frac{100}{7^3} \right\rfloor \]

Calculating each term:

1. \( \left\lfloor \frac{100}{7} \right\rfloor = \left\lfloor 14.2857 \right\rfloor = 14 \)
2. \( \left\lfloor \frac{100}{49} \right\rfloor = \left\lfloor 2.0408 \right\rfloor = 2 \)
3. \( \left\lfloor \frac{100}{343} \right\rfloor = \left\lfloor 0.2915 \right\rfloor = 0 \)

Adding these together gives:

\[ \text{Exponent of } 7 \text{ in } 100! = 14 + 2 + 0 = 16 \]

### Step 2: Calculate the exponent of 7 in \( 50! \)

Now, we do the same for \( n = 50 \):

\[ \text{Exponent of } 7 \text{ in } 50! = \left\lfloor \frac{50}{7} \right\rfloor + \left\lfloor \frac{50}{7^2} \right\rfloor \]

Calculating each term:

1. \( \left\lfloor \frac{50}{7} \right\rfloor = \left\lfloor 7.1428 \right\rfloor = 7 \)
2. \( \left\lfloor \frac{50}{49} \right\rfloor = \left\lfloor 1.0204 \right\rfloor = 1 \)

Adding these together gives:

\[ \text{Exponent of } 7 \text{ in } 50! = 7 + 1 = 8 \]

### Step 3: Combine the results

Now we can find the exponent of 7 in \( \binom{100}{50} \):

\[ \text{Exponent of } 7 \text{ in } \binom{100}{50} = \text{Exponent of } 7 \text{ in } 100! - 2 \times \text{Exponent of } 7 \text{ in } 50! \]

Substituting the values we found:

\[ \text{Exponent of } 7 \text{ in } \binom{100}{50} = 16 - 2 \times 8 = 16 - 16 = 0 \]

### Final Answer

The exponent of 7 in \( \binom{100}{50} \) is \( 0 \).

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