Respuesta :
Option C
2100 different committees that can be formed from 5 teachers and 10 students
Solution:
A school committee consists of 2 teachers and 4 students
The number of different committees that can be formed from 5 teachers and 10 students is to be found
This is a combination problem
A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected
The formula to calculate combinations is:
[tex]n C_{r}=\frac{n !}{r !(n-r) !}[/tex]
where n represents the number of items, and r represents the number of items being chosen at a time
Here we have to choose 2 teacher from 5 teacher and 4 student from 10 student
Hence we get,
Number of committees = [tex]5C_2 \times 10C_4[/tex]
[tex]\begin{aligned}&5 C_{2} \times 10 C_{4}=\frac{5 !}{2 !(5-2) !} \times \frac{10 !}{4 !(10-4) !}\\\\&5 C_{2} \times 10 C_{4}=\frac{5 !}{2 ! \times 3 !} \times \frac{10 !}{4 ! \times 6 !}\end{aligned}[/tex]
To get the factorial of a number n the given formula is used,
[tex]n !=n \times(n-1) \times(n-2) \ldots \times 2 \times 1[/tex]
Therefore,
[tex]5 C_{2} \times 10 C_{4}=\frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 3 \times 2 \times 1} \times \frac{10 \times 9 \times \ldots . \times 2 \times 1}{4 \times 3 \times 2 \times 1 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}[/tex]
[tex]5 C_{2} \times 10 C_{4}=2100[/tex]
Thus 2100 different committees that can be formed from 5 teachers and 10 students
