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At Flo’s Pancake House, pancakes can be ordered with any of

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At Flo’s Pancake House, pancakes can be ordered with any of [#permalink] New post 06 Jun 2017, 06:26
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At Flo’s Pancake House, pancakes can be ordered with any of six possible toppings. If no toppings were repeated, how many different ways are there to order pancakes with three toppings?

A) 20

B) 40

C) 54

D) 120

E) 720
[Reveal] Spoiler: OA

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Re: At Flo’s Pancake House, pancakes can be ordered with any of [#permalink] New post 06 Jun 2017, 15:29
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Carcass wrote:


At Flo’s Pancake House, pancakes can be ordered with any of six possible toppings. If no toppings were repeated, how many different ways are there to order pancakes with three toppings?

A) 20

B) 40

C) 54

D) 120

E) 720


The order in which we select the toppings does not matter, so we can use combinations.

We can select 3 toppings from 6 toppings in 6C3 ways

6C3 = (6)(5)(4)/(3)(2)(1)
= 20

Answer:
[Reveal] Spoiler:
A


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Brent
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Re: At Flo’s Pancake House, pancakes can be ordered with any of [#permalink] New post 29 Oct 2018, 16:52
How would this be different if you were allowed to repeat toppings?
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Re: At Flo’s Pancake House, pancakes can be ordered with any of [#permalink] New post 29 Oct 2018, 17:28
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msawicka wrote:
How would this be different if you were allowed to repeat toppings?



To the same question without repeating toppings:

Topping 1: 6 choices

Topping 2: 5 choices (6-1 toppings as one topping is already selected)

Topping 3: 4 choices (6-2 toppings as two toppings are already selected)

Total= \(6 \times 5 \times 4 = 120\)

Now say the three toppings are jelly, jam and nuts.

So 120 will contain {jelly, jam, nut},{nut,jelly, jam} ..... and 4 other combinations. Numbers of ways of arranging 3 things \(3!=6\)

So the correct number of options for pancakes =\(\frac{120}{6}=20\).

For repeating allowed you need to you need to add two more cases

2 toppings are same one different

So you have to choose 2 toppings and the order does not matter= \(\frac{6 \times 5}{2!}=15\) (\(C^6_{2}\))
Here you can repeat the first topping 2 times or the second topping twice so

Total choices= \(15 \times 2\)


All 3 toppings same

You have to choose 1 topping = \(6\) (\(C^6_{1}\))

Add them all up \(20 +15 + 6 =41\).
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Re: At Flo’s Pancake House, pancakes can be ordered with any of [#permalink] New post 29 Oct 2018, 21:41
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msawicka wrote:
How would this be different if you were allowed to repeat toppings?



Hi..

if we have no repetitions it is same as choosing 3 out of 6 so 6C3=\(\frac{6!}{3!3!}=\frac{6*5*4}{3*2}=20\)

Now when repetitions are allowed, following will get added..
1) Any one repeated twice..
so choose two out of 6 = 6C2 = \(\frac{6!}{4!2!}=15\)
But each choice will have two ways as one of them can be taken twice....
for example if two choosen is A and B... this will have two ways AAB or ABB
thus total = 15*2=30
2) any one repeated thrice so 6 ways

total = \(20+30+6=56\)
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: At Flo’s Pancake House, pancakes can be ordered with any of   [#permalink] 29 Oct 2018, 21:41
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