EDIT by Praet: thanks mayur,gmatblast. great work. i request other members to solve these tough problems. Dont always look for gmat type problems. the concepts are more important than gmat type Q's. thanks ndidi for posting the question.


Find the number of (a) combinations and (b) permutations of 4 letters each that can be made from the letters of the word TENNESSEE.

You are asking to compute combinations based on a multi-set {1.T, 4.E, 2.N, 2.S}, ill-formed question!

You framed this question, didn't you?

Nope, I did not frame the question. I found it in a Schaum's Outline algebra book and I couldn't solve it. It wasn't solved in the book either but the answer is listed.

The solution is this: -
There are 9 letters of 4 different sorts: -
TENNESSEE
T – 1; E – 4; N – 2; S – 2

Possibilities: -
Selections: -
1. All the 4 alike
2. 3 Alike, 1 different
3. 2 Alike, 2 Others alike
4. 2 Alike, 2 Others different
5. All the 4 different

Combinations: -

1. There is 1 group that has all the four alike and this can be selected in 1 way. (All Es)
2. 3 Alike can be selected in C(4,3) = 4 Ways (From the 4 Es) and one different can be selected in C(3,1) = 3 ways (from T, N & S). So, 4 X 3 =12 Ways.
3. 2 Alike and 2 others alike can be selected in C(3,2) = 3 Ways (E, N & S)
4. 2 Alike in C(3,1) = 3 Ways and then 2 others from the remaining 3 i.e., C(3,2) = 3 ways. So, 3 X3 = 9 Ways
5. All the 4 different in C(4,4) = 1 way

Total = 1 + 12 + 3 + 9 + 1 = 26 ways

Permutations: -

1. 1 X 4! / 4! = 1 way
2. 12 X 4! / 3! = 48 Ways
3. 3 X 4! / (2! X 2!) = 12 Ways
4. 9 X 4! / 2! = 108 Ways
5. 1 X 4! = 2 Ways

Total = 1 + 48 + 12 + 108 + 2 = 171 Ways
_________________

Mayur

gmatblast,
In fact, I goofed up. I agree that the total no. of selections is 17 as follows: -
There are 9 letters of 4 different sorts: -
TENNESSEE
T – 1; E – 4; N – 2; S – 2

Possibilities: -
Selections: -
1. All the 4 alike
2. 3 Alike, 1 different
3. 2 Alike, 2 Others alike
4. 2 Alike, 2 Others different
5. All the 4 different

Combinations: -

1. There is 1 group that has all the four alike and this can be selected in 1 way. (All Es)
2. 3 Alike can be selected in 1 Way (From the 4 Es) and one different can be selected in C(3,1) = 3 ways (from T, N & S). So, 1 X 3 =3 Ways.
3. 2 Alike and 2 others alike can be selected in C(3,2) = 3 Ways (E, N & S)
4. 2 Alike in C(3,1) = 3 Ways and then 2 others from the remaining 3 i.e., C(3,2) = 3 ways. So, 3 X3 = 9 Ways
5. All the 4 different in C(4,4) = 1 way

Total = 1 + 3 + 3 + 9 + 1 = 17 ways

Permutations: -

1. 1 X 4! / 4! = 1 way
2. 3 X 4! / 3! = 12 Ways
3. 3 X 4! / (2! X 2!) = 18 Ways
4. 9 X 4! / 2! = 108 Ways
5. 1 X 4! = 24 Ways

Total = 1 + 12 + 18 + 108 + 24 = 163 Ways


The problem asks for permutations of 4 lettered words. Hence, we have to use the above method. The solution you gave is for the permutation of all the letters, i.e., 9 letters.


In fact, I posted a similar problem with the word "PROPORTION" on May 18 and posted the answer on May 19 2004.


ndidi204,
Please let us know the answer.
_________________

Mayur

Mayur, you got it right!!!

# of Combinations = 17
# of Permutations = 163

Good job.

I am getting a different answer for the permutations. I would appreciate some clarifications

chosing 4 letters from TENNESSEE and arranging them can be done in following ways
TEEE = 4!/3! = 4
TEEN = 4!/2! = 12
TEES = 4!/2! = 12
TENN = 4!/2! = 12
TESS = 4!/2! = 12
TENS = 4! = 24
TNNS = 4!/2! = 12
TNSS = 4!/2! = 12
EEEE = 4!/4! = 1
EEEN = 4!/3! = 4
EEES = 4!/3! = 4
EENN = 4!/( 2! 2! ) = 6
EESS = 4!/(2! 2!) = 6
EENS = 4!/2! = 12
NNSS = 4!/(2! 2!) = 6

Total = 139

Yeah you are right. Brute force is not a good method to solve these problems.

139+12+12 = 163

few minutes two late staring at these from work
_________________

Best Regards,

Paul

When we select (combinations), we arrange(permutations) them. In the first case, we select all the 4 alike in 1 way and arrange them 4!/4! ways.

Similarly, in the second case, we select in 12 ways and arrange the letters of the word in 4!/3! ways.