Total number of ways of choosing one hundred students from the four hundred applicants for accommodation.?

Question

Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice.

Answer 1

Since you did not specify how many students are incompatible with how many other students, I can't answer that part of the question. However, I can tell you the number of unique ways to choose 100 students (in no particular order) from a group of 400.

We use the combination function 400-choose-100. Remember that "!" indicates the factorial operation.

com(400,100)
= 400! / (100! * (400-100)!)
= 400! / (100! * 300!)
= 2241854791554337561
92321038720169855484
54111774762959903999
42258896013007429693
894018935107174320

(that's all one number).

The number is very, very large. Written in scientific notation, it would look like this:

2.242 * 10^96

Answer 2

Four small books of 100 duplicate tickets (so that you don't have to count 400 pieces of paper). Names on each stub to include all 400, into a hat, and pick out 100. Keep your list of 'not to appear in final choice" handy, and, if any of those names is pulled from the hat, discard it and on to another ticket! Don't use the Maths - that's too hard!

Answer 3

C(400, 100) = 400!/(300!100!)

Answer 4

This is a combination
C(n,r) = n! / [(n-r)!r!]

c(400,100) = 400! / [(400-100)!100!]

= 400! /[300!(100!)]

= an incredibly large number...........
= 2.24186(10^96)

Answer 5

400C100 = 400!/(400-100)! 100!
= 400!/300! * 100!
= 400*399*398.......*301/100!