Given 10 chips each of them different colors including red a

Given 10 chips each of them different colors (including red and white). For each of the following questions explain your answer. You should use factorial, combination, permutation, or simple products whener it is possible and you do not need to give the number it is equal to. How many different ways can we put:

a) 6 of the 10 chips in a line?

b) 6 of the 10 chips in a line s.t. the red chip must be used?

c) 6 of the 10 chips in a line s.t. the red chip is used and the white is not?

d) 6 of the 10 chips in a line s.t. both the red and white chip is used?

e) 6 of the 10 chips in a line s.t. both the red and white chip is used and they are next to each other?

For a, I tried: 10!/4! and for b) I tried 10!/4! - 9!/3!.

Solution

a)the first one is 10P6 =10!/(10-6)! =10!/4!

b)red chip must be used=all permutation-red chip not used

                                   =10!/4! -9!/3!

c)red and white chip is used=total permuatation-(redchip,white chip not used)

                                       =10!/4! -8!/2!

d)both red chip and white chip are next to each other is ,total item =9 as 2 unit(red+white) as a single unit

total permutation- (1 not used)

10!/4! -9!/3!