# 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...

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.

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**There are 6 possibilities that diet pop can be in five cups.**

**1. No diet pop in any cup**

Therefore she would write,

rrrrr

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)

`= 1`

**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!)`

`= 5`

**3. Only two cups of diet pop**

`n_2 = (5!)/(2!xx3!)`

`= 10`

4. Only three cups of diet pop

`n_3 = (5!)/(3! xx 2!)`

`= 10`

**5. Only four cups of diet pop**

n_4 = (5!)/(4! xx 1!)

= 5

**6. All five cups are diet pop**

`n_5 = (5!)/(5!)`

`= 1`

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.**