Find the exponent of 10 in 75C25.

To find the exponent of 10 in the binomial coefficient \(75C25\), we need to determine how many times 10 divides into this value. Since 10 can be factored into its prime components, 2 and 5, we will find the minimum of the counts of these two prime factors in \(75C25\).

Understanding the Binomial Coefficient

The binomial coefficient \(nCk\) is calculated using the formula:

\(nCk = \frac{n!}{k!(n-k)!}\)

In our case, \(n = 75\) and \(k = 25\), so:

\(75C25 = \frac{75!}{25! \cdot 50!}\)

Finding the Prime Factorization

To find the exponent of 10, we need to count the factors of 2 and 5 in \(75C25\). The exponent of 10 will be the smaller of these two counts.

Counting Factors of 5

We can find the number of times a prime \(p\) divides \(n!\) using the formula:

\( \text{Count}(p, n!) = \sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor \)

Applying this for \(p = 5\):

• For \(75!\): \( \left\lfloor \frac{75}{5} \right\rfloor + \left\lfloor \frac{75}{25} \right\rfloor = 15 + 3 = 18\)
• For \(25!\): \( \left\lfloor \frac{25}{5} \right\rfloor + \left\lfloor \frac{25}{25} \right\rfloor = 5 + 1 = 6\)
• For \(50!\): \( \left\lfloor \frac{50}{5} \right\rfloor + \left\lfloor \frac{50}{25} \right\rfloor = 10 + 2 = 12\)

Now, we can calculate the total count of factors of 5 in \(75C25\):

\( \text{Count of 5} = 18 - (6 + 12) = 0\)

Counting Factors of 2

Next, we do the same for \(p = 2\):

• For \(75!\): \( \left\lfloor \frac{75}{2} \right\rfloor + \left\lfloor \frac{75}{4} \right\rfloor + \left\lfloor \frac{75}{8} \right\rfloor + \left\lfloor \frac{75}{16} \right\rfloor + \left\lfloor \frac{75}{32} \right\rfloor + \left\lfloor \frac{75}{64} \right\rfloor = 37 + 18 + 9 + 4 + 2 + 1 = 71\)
• For \(25!\): \( \left\lfloor \frac{25}{2} \right\rfloor + \left\lfloor \frac{25}{4} \right\rfloor + \left\lfloor \frac{25}{8} \right\rfloor + \left\lfloor \frac{25}{16} \right\rfloor = 12 + 6 + 3 + 1 = 22\)
• For \(50!\): \( \left\lfloor \frac{50}{2} \right\rfloor + \left\lfloor \frac{50}{4} \right\rfloor + \left\lfloor \frac{50}{8} \right\rfloor + \left\lfloor \frac{50}{16} \right\rfloor + \left\lfloor \frac{50}{32} \right\rfloor = 25 + 12 + 6 + 3 + 1 = 47\)

Now, we calculate the total count of factors of 2 in \(75C25\):

\( \text{Count of 2} = 71 - (22 + 47) = 2\)

Final Calculation

Now that we have both counts:

• Count of 5: 0
• Count of 2: 2

The exponent of 10 in \(75C25\) is the minimum of these two counts:

\( \text{Exponent of 10} = \min(0, 2) = 0\)

Thus, the exponent of 10 in \(75C25\) is 0, meaning that \(75C25\) is not divisible by 10 at all.