Find the value of following.
(i) `""^(15)C_(11)` (ii) `""^(10)C_(4)`

Answer

To solve the problem, we need to calculate the values of \( \binom{15}{11} \) and \( \binom{10}{4} \) using the formula for combinations:

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

### Step 1: Calculate \( \binom{15}{11} \)

Using the formula for combinations:

\[ \binom{15}{11} = \frac{15!}{11!(15-11)!} = \frac{15!}{11! \cdot 4!} \]

### Step 2: Simplify \( \binom{15}{11} \)

We can simplify \( 15! \) as follows:

\[ 15! = 15 \times 14 \times 13 \times 12 \times 11! \]

Substituting this into our equation gives:

\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12 \times 11!}{11! \cdot 4!} \]

### Step 3: Cancel \( 11! \)

The \( 11! \) in the numerator and denominator cancels out:

\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12}{4!} \]

### Step 4: Calculate \( 4! \)

Now, we need to calculate \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 5: Substitute \( 4! \) into the equation

Now substitute \( 4! \) back into the equation:

\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12}{24} \]

### Step 6: Calculate the numerator

Calculating the numerator:

\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]

### Step 7: Divide by \( 24 \)

Now divide the result by \( 24 \):

\[ \binom{15}{11} = \frac{32760}{24} = 1365 \]

### Step 8: Calculate \( \binom{10}{4} \)

Now we calculate \( \binom{10}{4} \):

\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \]

### Step 9: Simplify \( \binom{10}{4} \)

We can simplify \( 10! \):

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6! \]

Substituting this gives:

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \cdot 6!} \]

### Step 10: Cancel \( 6! \)

The \( 6! \) cancels out:

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4!} \]

### Step 11: Substitute \( 4! \)

Using \( 4! = 24 \):

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{24} \]

### Step 12: Calculate the numerator

Calculating the numerator:

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]

### Step 13: Divide by \( 24 \)

Now divide by \( 24 \):

\[ \binom{10}{4} = \frac{5040}{24} = 210 \]

### Final Answers

Thus, the final answers are:

(i) \( \binom{15}{11} = 1365 \)

(ii) \( \binom{10}{4} = 210 \)

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