To determine the number of ways the first six customers can be chosen out of thirty-six, we need to calculate the number of combinations possible. When choosing combinations, the order in which the customers are chosen does not matter.
The general formula for combinations [tex]\(_nC_r\)[/tex] is given by:
[tex]\[ _nC_r = \frac{n!}{r!(n-r)!} \][/tex]
where [tex]\( n \)[/tex] is the total number of items, [tex]\( r \)[/tex] is the number of items to choose, and [tex]\( ! \)[/tex] denotes factorial (the product of all positive integers up to that number).
In this case:
- [tex]\( n = 36 \)[/tex]: the total number of customers
- [tex]\( r = 6 \)[/tex]: the number of customers to be chosen
Using these values, the problem can be translated to finding [tex]\( _{36}C_{6} \)[/tex]:
[tex]\[ _{36}C_{6} = \frac{36!}{6!(36-6)!} \][/tex]
[tex]\[ _{36}C_{6} = \frac{36!}{6! \times 30!} \][/tex]
Now, calculating factorials for large numbers is complex and often best done with a calculator or a program for efficiency. According to the problem, the result of this calculation is:
[tex]\[ _{36}C_{6} = 1,947,792 \][/tex]
This value represents the number of distinct ways we can choose 6 customers from a group of 36 without regard to the order of selection.
Therefore, the correct answer is:
1,947,792
