These questions are really beautiful. It would be nice if you could refer the name of the book or source. I definitely dont doubt the credibility of the question, these question can appear on the actual GMAT. These questions are GOLDMINE!!!!
ENTRANCE
1) The first problem is of combination.
4 letters are to be chosen out of 8 letters but 2 letters are in repetition.
There are 6 distinct letters. Lets take this case by case.
I......... 4 letters are chosen from 6 distinct letters.
6C4 = 15 ways
II........E & N are in repetition therefore only way we can choose both of them together i.e. NN EE is 1 way
III......No. of ways 2 EE's are chosen in 4 letter is 1*5C2 = 10 ways
(1*5C2 means we can put 2 E in either of the 2 letters of 4 letters and remaining we are left with 2 more letters to be chosen out of 5 distinct letters, therefore 5C2)
IV......No. of ways 2 NN's are chosen in 4 letter is 1*5C2 = 10 ways
Therefore total number of selections is 15+1+10+10= 36
2) The second question is bit tricky. If you understand the anagram form this can be a bit easier to understand.
I.....lets take only distinct letters i.e. ENTRAC (6 letters)
These can be arranged in 6*5*4*3 or 6P4 = 360 ways.
II....lets take the repetitions EENN
These can be arranged in 4!/(2!2!) = 6 ways
III...lets take condition only EE and TRANC (1 repetition + 5 distinct letters)
12/2! * 5*4 = 120 ways or 12!/2*5P2
12 because in first two places we can arrange one E in 4 ways and the other E in 3 ways but since we cannot differentiate between the two EE we have to divide by 2. IF WE DONT DIVIDE BY 2 THE METHOD WILL DIFFERENTIATE E1 E2 T R & E2 E1 T R which we dont want.
IV....Condition NN and TRACE (Reasoning same as above)
12!/2 * 5*4 = 120 ways or 12!/2*5P2 = 120 ways
Add all of them 360+6+120+120 = 606 arrangements.
Another way of looking at the same problem:
How many 4 digit numbers can formed using letters 1,2,2,3,3,4,5,6.
It took me more than half hour to do these, but I still believe that these are one of the best questions I have ever solved for permutation & combination.
