To evaluate the expression [tex]\(\frac{13!}{11! \cdot 2!}\)[/tex], let's break down the calculation step-by-step.
First, recall that the factorial of a number [tex]\(n\)[/tex], denoted as [tex]\(n!\)[/tex], is the product of all positive integers up to [tex]\(n\)[/tex]. Therefore:
[tex]\[ 13! = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 2! = 2 \times 1 \][/tex]
Now, we observe that [tex]\(11!\)[/tex] is a common factor both in the numerator and the denominator which allows us to simplify the fraction. We can write:
[tex]\[ 13! = 13 \times 12 \times 11! \][/tex]
Thus, the expression becomes:
[tex]\[ \frac{13!}{11! \cdot 2!} = \frac{13 \times 12 \times 11!}{11! \cdot 2!} \][/tex]
Since [tex]\(11!\)[/tex] appears in both the numerator and denominator, they cancel each other out, leaving:
[tex]\[ \frac{13 \times 12 \times 11!}{11! \cdot 2!} = \frac{13 \times 12}{2!} = \frac{13 \times 12}{2 \times 1} \][/tex]
Now we compute the remaining multiplication and division:
[tex]\[ 13 \times 12 = 156 \][/tex]
[tex]\[ 2 \times 1 = 2 \][/tex]
So,
[tex]\[ \frac{156}{2} = 78 \][/tex]
Therefore, the simplified value of the expression [tex]\(\frac{13!}{11! \cdot 2!}\)[/tex] is:
[tex]\[ 78 \][/tex]
So, [tex]\(\frac{13!}{11!2!} = 78\)[/tex]
