Sure, let's calculate [tex]\( { }_{35} C_{12} \)[/tex] using the combination formula given:
[tex]\[
{}_{n}C_r = \frac{n!}{r!(n-r)!}
\][/tex]
Here, [tex]\( n = 35 \)[/tex] and [tex]\( r = 12 \)[/tex].
### Step 1: Calculate the factorial of [tex]\( n \)[/tex]
We need to find [tex]\( 35! \)[/tex]:
[tex]\[
35! = 10333147966386144929666651337523200000000
\][/tex]
### Step 2: Calculate the factorial of [tex]\( r \)[/tex]
Next, we calculate [tex]\( 12! \)[/tex]:
[tex]\[
12! = 479001600
\][/tex]
### Step 3: Calculate the factorial of [tex]\((n - r)\)[/tex]
We also need to find [tex]\( (35 - 12)! \)[/tex], which is [tex]\( 23! \)[/tex]:
[tex]\[
23! = 25852016738884976640000
\][/tex]
### Step 4: Plug these values into the combination formula
Now we can substitute these values into the formula:
[tex]\[
{}_{35}C_{12} = \frac{35!}{12!(35-12)!} = \frac{35!}{12! \times 23!}
\][/tex]
[tex]\[
{}_{35}C_{12} = \frac{10333147966386144929666651337523200000000}{479001600 \times 25852016738884976640000}
\][/tex]
### Step 5: Simplify the expression
Performing the division gives us the final result:
[tex]\[
{}_{35}C_{12} = 834451800.0
\][/tex]
So, the value of [tex]\( { }_{35}C_{12} \)[/tex] is [tex]\( 834451800 \)[/tex].
