GMAT PS
How many four-digit positive integers can be formed by using the digits from 1 to 9 so that two digits are equal to each other and the remaining two are also equal to each other but different from the other two ?
A. 400
B. 1728
C. 108
D. 216
E. 432
I am getting answer as 432, but the OA is D. Here is the approach I used.
First number can be selected from 1 to 9 – in 9 ways, 2nd number has to be same as the one selected so only 1 way, Third number can be selected from remaining 8 numbers in 8 ways and 4th number again has to be only 1
so in total (9*1*8*1)*4!/2!*2! = 72*6 = 432 Can anyone please explain what mistake I am making here?
If i use 9C2 * (4!/(2!*2!) i get correct answer, which i agree is also one of the solution But unable to understand what is wrong with my first approach.
Thanks,
Nitesh
3 possible combination as below
XXYY = 9*1*8*1
XYXY = 9*8*1*1
XYYX =9*8*1*1
= 9*1*8*1 + 9*8*1*1 + 9*8*1*1 = 9*8*3
= 216
In your approach you miss one duplicate i believe.
XXYY = YYXX
XYXY = YXYX
XYYX = YXXY
which suppose to be = 72*6/2 = 72* 3 = 216
Perfect explanation by priyanshu.
@ Nitesh
Just to clarify,
In your approach, I think, you are fixing it as an order, then again doing the arrangement, that is where the confusion. So you are getting the wrong answer.
First number can be selected from 1 to 9 – in 9 ways, 2nd number has to be same as the one selected so only 1 way, Third number can be selected from remaining 8 numbers in 8 ways and 4th number again has to be only 1
so in total (9*1*8*1)*4!/2!*2! = 72*6 = 432 Can anyone please explain what mistake I am making here?
In this first number to select there 9 ways,
You can place that, first digit select in three different ways again.
Like suppose, pick here say 5 as the first digit,
then it could be,
5 _ _ 5
5 _ 5 _
5 5 _ _
So, the number of ways of picking the first digit is 9 and this could be arranged in three ways . So, 9 * 3 and also the second digit could be picked in 8 ways.
So, the answer is 9 * 3 * 8 = 216.
Hope this helps.
Since, you have already mentioned you got the second approach. I leave it to you.
Let us know if you have any queries.
