Sammy has flavors of candies with which to make goody bags

Sammy has x flavors of candies with which to make goody bags for Franks birthday party. Sammy tosses out y flavors, because he doesnt like them. How many different 10-flavor bags can Sammy make from the remaining flavors? (It doesnt matter how many candies are in a bag, only how many flavors).

In order to calculate how many 10-flavor bags can Sammy make from the remaining (x-y) flavors, we should know the value of x-y. The answer would simply be \(C^{10}_{x-y}\). For example if he has 11 flavors (if x-y=11), then he can make \(C^{10}_{11}=11\) different 10-flavor bags.

(1) If Sammy had thrown away 2 additional flavors of candy, he could have made exactly 3,003 different 10-flavor bags. We are told that \(C^{10}_n=3,003\), where \(n=(x-y)-2\): he can make 3,003 10-flavor bags out of n flavors. Now, n can take only one particular value, so we can find n (it really doesn't matter what is the value n, important is that we can find it), hence we can find the value of x-y (x-y=n+2). Sufficient.

(2) x = y + 17 --> x-y=17. Directly gives us the value of x-y. Sufficient.

Answer: D.

Hope it's clear.
_________________

Hey thanx.
But I shall like to bother you again.
I have come across around 3 questions of this type where the value of the total, of which we are supposed to make selection, is not known. Even in this question also, I knew that there could be only one value for which the answer comes out to be 3003, but since I was unable to come out with the solution, I sought help.
So my question is that can't there be any more value for n for which nC10 is 3003?

This is explained here: how-many-different-5-person-teams-can-be-formed-from-a-group-96244.html#p741834 and here: how-many-different-5-person-teams-can-be-formed-from-a-group-96244.html#p742026

Similar questions to practice:
a-certain-panel-is-to-be-composed-of-exactly-three-women-and-108964.html
a-box-contains-10-light-bulbs-fewer-than-half-of-which-are-99940.html
how-many-different-5-person-teams-can-be-formed-from-a-group-96244.html

Hope it helps.
_________________

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE


_________________

OK, let me try this one

Question stem basically says we have x-y flavors and we need to pick 10 out of these

How many combinations can we make?

Statement 1

So this is giving us the number of combinations of x-y-2, therefore we can imply what x-y is. Do we need the calculation? No, there will only be one number that will give this answer when deciding to pick 10 out of it.

Suff

Statement 2

Rearranging x-y = 17

We have the total

Suff

Answer is D

Hope it helps

Cheers!
J

I was concerned more with finding the value of y..but looks like it was not really necessary...
how i approached the question:
1. (x-y-2)C10 = 3003 -> find prime factorization of 3003...we can see that we have 3*7*11*13...
(x-y-2)x(x-y-1)x(x-y)(x-y+1) x etc. / 10!*(x-y-12)!

how can we solve further???

2. x-y=17
17C10 - we can find the answer...
so somewhere I did not see how to solve A...

Can anyone help how to solve the combination:
(x-y-2)C10=3003

We need to determine the number of different 10-flavor goody bags Sammy can make for Frank. Since there were x flavors to begin with and y were thrown away, there are (x - y) remaining flavors. Moreover, since the order in which the flavors are chosen is not important, this is a combination problem. In other words, we must determine whether we have enough information to calculate (x - y)C10.

Statement One Alone:

If Sammy had thrown away 2 additional flavors of candy, he could have made exactly 3,003 different 10-flavor bags.

This statement tells us that (x - y b- 2)C10 = 3003. The important thing to remember when approaching questions like this is that we don’t actually need to find the answer; we simply need to determine whether we have enough information to answer the question. In this case, since there is a unique value for (x - y - 2) that satisfies (x - y - 2)C10 = (x - y - 2)!/((x - y - 12)! * 10!), we can determine (x - y - 2), and thus we can determine x - y. After x - y is determined, it is easy to calculate (x - y)C10.

Statement one alone provides enough information to answer the question.

Statement Two Alone:

x = y + 17

We have x - y = 17; thus, 17C10 = 17!/(7! * 10!) can be calculated.

Statement two alone provides enough information to answer the question.

Answer: D
_________________

How do you make sure that it has only one solution without solving it, did we just assume it?

It may have 10 different roots. (let's say x-y-2 is a) C(a, 10) = ((a).(a-1).(a-2)....(a-9))/10!=3003

[quote="financestudent"]How do you make sure that it has only one solution without solving it, did we just assume it?

It may have 10 different roots. (let's say x-y-2 is a) C(a, 10) = ((a).(a-1).(a-2)....(a-9))/10!=3003


Hello

You are right that we can simplify this as: ((a).(a-1).(a-2)....(a-9))/10!=3003
We can say that ((a).(a-1).(a-2)....(a-9)) = 3003*10!

Now since 3003*10! is a unique number (no matter how big it is), there can be ONLY single value of a positive integer 'a' such that ((a).(a-1).(a-2)....(a-9)) equals 3003*10!. Since this is data sufficiency, we dont need to find the value of 'a'. If we know for sure that a unique value of 'a' exists, the statement is sufficient.

Hi there, thank you for the explanation.

Can you also explain how you figured out that 3003*10! is a unique number or how can I tell that a number is unique (especially like this big one)?

Bunuel amanvermagmat

(1) If Sammy had thrown away 2 additional flavors of candy, he could have made exactly 3,003 different 10-flavor bags.
From the question, \((x-y-2)C_{10}\) OR
\(nC_{10}\) = 3003
\(\frac{n!}{10!* (n-10)!}\) = 3003
by trial we can get n= 17
sufficient

(2) x = y + 17
x-y = 17
sufficient

option D

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________