To determine the number of ways the first 4 guests can arrive at a party if 15 guests have been invited, we will use the concept of permutations. Permutations are used when the order of selection matters.
The formula for permutations is given by:
[tex]\[
{}_nP_r = \frac{n!}{(n-r)!}
\][/tex]
Here, [tex]\( n \)[/tex] is the total number of items (or people, in this case), and [tex]\( r \)[/tex] is the number of items (or people) to choose.
Given:
[tex]\( n = 15 \)[/tex] (total guests)
[tex]\( r = 4 \)[/tex] (guests to choose)
Using the permutation formula:
[tex]\[
{}_{15}P_4 = \frac{15!}{(15-4)!} = \frac{15!}{11!}
\][/tex]
When we expand this, we compute:
[tex]\[
\frac{15!}{11!} = \frac{15 \times 14 \times 13 \times 12 \times 11!}{11!}
\][/tex]
The [tex]\(11!\)[/tex] in the numerator and denominator cancels out, leaving us with:
[tex]\[
15 \times 14 \times 13 \times 12
\][/tex]
Multiplying these values together will give us the number of permutations:
[tex]\[
15 \times 14 \times 13 \times 12 = 32760
\][/tex]
Thus, the number of ways the first 4 guests can arrive at the party is:
[tex]\[
32760 \, \text{ways}
\][/tex]
