Evaluate
` C(31,26)-C(30,26)`

Answer

To evaluate \( C(31, 26) - C(30, 26) \), we can use the property of combinations which states:

\[ C(n, r) = C(n-1, r) + C(n-1, r-1) \]

This property can be rearranged to express \( C(n, r) \) in terms of two other combinations. Specifically, we can say:

\[ C(31, 26) = C(30, 26) + C(30, 25) \]

Now, substituting this into our original expression:

\[ C(31, 26) - C(30, 26) = (C(30, 26) + C(30, 25)) - C(30, 26) \]

This simplifies to:

\[ C(31, 26) - C(30, 26) = C(30, 25) \]

Next, we need to calculate \( C(30, 25) \). The formula for combinations is given by:

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

Applying this to \( C(30, 25) \):

\[ C(30, 25) = \frac{30!}{25! \cdot (30 - 25)!} = \frac{30!}{25! \cdot 5!} \]

Now, we can simplify \( C(30, 25) \):

\[ C(30, 25) = \frac{30 \times 29 \times 28 \times 27 \times 26}{5 \times 4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 30 \times 29 = 870 \]
\[ 870 \times 28 = 24360 \]
\[ 24360 \times 27 = 657720 \]
\[ 657720 \times 26 = 17153220 \]

Now, calculating the denominator:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Now we divide the numerator by the denominator:

\[ C(30, 25) = \frac{17153136}{120} = 142506 \]

Thus, the final answer is:

\[ C(31, 26) - C(30, 26) = 142506 \]