Sure, let's solve the given expression step-by-step:
The expression we need to evaluate is:
[tex]\[
\frac{\frac{n!}{r!(n-r)!}}{1-25-26-31-6s-2}
\][/tex]
### Step 1: Calculate [tex]\( n! \)[/tex]
Given [tex]\( n = 5 \)[/tex]:
[tex]\[
n! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
### Step 2: Calculate [tex]\( r! \)[/tex]
Given [tex]\( r = 3 \)[/tex]:
[tex]\[
r! = 3! = 3 \times 2 \times 1 = 6
\][/tex]
### Step 3: Calculate [tex]\( (n - r)! \)[/tex]
Given [tex]\( n = 5 \)[/tex] and [tex]\( r = 3 \)[/tex]:
[tex]\[
(n - r)! = (5 - 3)! = 2! = 2 \times 1 = 2
\][/tex]
### Step 4: Calculate the numerator [tex]\( \frac{n!}{r!(n-r)!} \)[/tex]
Now, substitute the values into the numerator formula:
[tex]\[
\frac{n!}{r!(n-r)!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10
\][/tex]
### Step 5: Calculate the denominator [tex]\( 1 - 25 - 26 - 31 - 6s - 2 \)[/tex]
Given [tex]\( s = 2 \)[/tex]:
[tex]\[
1 - 25 - 26 - 31 - 6s - 2 = 1 - 25 - 26 - 31 - 6 \times 2 - 2 = 1 - 25 - 26 - 31 - 12 - 2 = 1 - 96 = -95
\][/tex]
### Step 6: Calculate the final result
Now we have the numerator value [tex]\( 10 \)[/tex] and the denominator value [tex]\( -95 \)[/tex]:
[tex]\[
\frac{10}{-95} = -\frac{10}{95} = -\frac{2}{19} \approx -0.10526315789473684
\][/tex]
So, the final detailed step-by-step result of the expression is approximately:
[tex]\[
-0.10526315789473684
\][/tex]
