S claims she can tell difference between diet pop and regular pop. you line up 5 cups(cup 1, cup2,cup3,cup4,cup5). she tastes each and records d or r.
How many different ways can the d's and r's be recorded if sandra guesses which cups have diet pop and which have regular.
There are 6 possibilities that diet pop can be in five cups.
1. No diet pop in any cup
Therefore she would write,
There is only one way of arranaging these 5 regular cups of pop. This can be calculated as below
`n_0 = (5!)/(5!)` (since there are five similar letters)
2. Only one cup of diet pop
The number of way you can arrange one cup of diet pop with four cups of regular pop = `n_2`
`n_1 = (5!)/(1!xx4!)`
3. Only two cups of diet pop
`n_2 = (5!)/(2!xx3!)`
4. Only three cups of diet pop
`n_3 = (5!)/(3! xx 2!)`
5. Only four cups of diet pop
n_4 = (5!)/(4! xx 1!)
6. All five cups are diet pop
`n_5 = (5!)/(5!)`
Therefore the total number of ways that d and r can be recorded is N
`N = n_0+n_1+n_2+n_3+n_4+n_5`
`N = 1+5+10+10+5+1`
`N = 32`
Therefore there are 32 ways that d and r can be recorded.