Can someone please help with this Grade 12 Data question?

Question

A teacher organizing a field trip finds that 50 students have signed up. However, the bus has only 47 seats, so a few students will have to travel by car. The teacher and one other supervisor must go on the bus. Explain two different methods for using combinations to find how each teacher can choose which students go on the bus. Show that both methods produce the same answer.

Thanks in Advance :)

Answer 1

i presume that there are only 2 supervisors ( the techer, plus one). they both have to travel by bus, therefor there will be 47-2=45 students in the bus.

that means that 3 students have to travel by car. now, you can think "how to choose the students to go on the bus, from the toal of 50, which is combination of 50 taken by 47, or you can think of how to choose the students that will go by car, which is combination of 50 taken by 3.  both of the methods produce the same results, because each time you choose a different student to go on the bus, the remainig set of students to go by car changes, and each time you choose a student to go by car, the remaining set of students to go by bus changes.

Combination of n taken by k = combination of n taken by

(n-k).

thats beacuse Combination by n taken by k = n! / [k! (n-k)!]

and by changing k with (n-k), in the above formula, we can obtain the formula for combination of  taken by (n-k), which is:

n! / (n-k)! [n-(n-k)] =  

n!/ (n-k)! ( n-n+k)! =

n! / (n-k)!k!

the same result as the combination of n taken by k :)

hope this helps!

good luck !