gmatbusters wrote:
How many different 4-letter words can be formed using the letters of the word CONSTITUTION such it starts with C and ends with N?
A) 30
B) 33
C) 40
D) 42
E) 24
The word "CONSTITUTION" has count of each letter as follows
C = 1
O = 2
N = 2
S = 1
T = 3
I = 2
U = 1
Now 4 letter words are to be formed, from these letters & each word has to be distinct. Also each letter has to start with C & end in N.
So we have format which is C _ _ N, hence we are left with all the above letters, except C & one N.
lets consider each of the remaining letters (O,N,S,T, I & U) in the second blank & calculate the possible arrangements for the third blank.
Case 1 - C O _ N
The blank can be filled with 5 letters (N, S, T, I & U). The letters in 2nd blank & 3rd blank can be arranged among themselves in 2 ways.
Hence total arrangements for Case 1 = 5 * 2 = 10 words
Case 2 - C N _ N
The blank can be filled with 4 letters (S, T, I & U), since arrangements with O are already considered in case 1, we neglect those.
The letters in 2nd blank & 3rd blank can be arranged among themselves in 2 ways.
Hence total arrangements for Case 2 = 4 * 2 = 8 words
Case 3 - C S _ N
The blank can be filled with 3 letters (T, I & U), since arrangements for O & N are already considered in Cases 1 & 2, we neglect those.
The letters in 2nd blank & 3rd blank can be arranged among themselves in 2 ways.
Hence total arrangements for Case 3 = 3 * 2 = 6 words
Case 4 - C T _ N
The blank can be filled with 2 letters ( I & U), since arrangements for O, N & S are already considered in Cases 1,2 & 3, we neglect those.
The letters in 2nd blank & 3rd blank can be arranged among themselves in 2 ways.
Hence total arrangements for Case 4 = 2 * 2 = 4 words
Case 5 - C I _ N
The blank can be filled with 1 letter (U), since arrangements for O, N, S & T are already considered in Cases 1,2,3 & 4, we neglect those.
The letters in 2nd blank & 3rd blank can be arranged among themselves in 2 ways.
Hence total arrangements for Case 5 = 1 * 2 = 2 words
Now Case 6 - When two same letters are selected
C O O N
C T T N
C I I N
Hence 3 arrangements
Hence total # of different words formed = 10 + 8 + 6 + 4 + 2 + 3 =33
Answer B.
I am sure
gmatbusters will come up with an even simpler solution than this.
Thanks,
GyM
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