Sid intended to type a seven-digit number, but the two 3's he meant to type did not appear. What appeared instead was the five-digit number 52115. How many different seven-digit numbers could Sid have meant to type?

A. 10
B. 16
C. 21
D. 24
E. 27


Kudos for a correct solution.
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The other numbers are fixed so we have to choose 2 spaces for the 2 numbers.

7C2=21


Answer : C

My idea was as follows, I'm not exactly great in these kind of questions so it would be nice if someone commented.
We got 2 numbers to place.
1st one can be placed in one of the 6 spots: _ X _ X _ X _ X _ X _ (6 spots = 6 underscores), we can do it in 6 different ways
2nd number can be placed in one of the 7 spots (after having placed the first number we got a 6 digit now: _ X _ X _ X _ X _ X _ X _ : 7 spots = 7 underscores), respectively 7 ways.
As a result we get 6*7 = 42 ways
Since we got duplicates coz of numbers being equal, we'll have to cut it in half, which gives us 42/2 = 21, the answer C.

We can choose ways to place two 3 three in a 7 digit integer for which 5 digits are fixed is by 7C2 ways.
7C2 = 21
Hence answer is C

MAGOOSH OFFICIAL SOLUTION:

First of all, notice that the numerals 5-2-1-1-5 must be in that order --- the order may be broken at any point but one or two 3's, but ignoring the 3's, those five must have that order.

As is often the case, in this counting problem we have a variety of ways to frame the question. I believe the most straightforward is as follows. Consider the seven "blank spaces" into which we will write the seven digits of the original number.

_ _ _ _ _ _ _

Now, select any two of those spaces, and put the two 3's there. The two 3's could be next to each other or apart. Suppose, for example, we put them here:

_ _ 3 _ _3 _

At this point, notice that everything else about the number is determined: we simply will put the digits 5-2-1-1-2 in order in the remaining spaces. Just picking two of the seven spaces for the location of the two 3's is enough to determine the entire original number. Well, there are 7C2 ways of selecting two slots from seven, so 7C2 must be the number of possible original numbers that Sid intended to write.

See this post for various techniques on calculating combinations.

7C2 = 21

So there are 21 possible original numbers that Sid could have intended.

Answer = C
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