How many four lettered words can be formed using the letters of the wo

Question

How many four lettered words can be formed using the letters of the word ARRANGE?

A. 120
B. 144
C. 264
D. 270
E. 290

Source: Indian B-school entrance exam

BrentGMATPrepNow · Accepted Answer

Unless I'm missing a super fast shortcut, I think this question is too time consuming to be an official GMAT question. 
The reason for this is that we need to consider for separate cases: 
- All 4 letters are different (5x4x3x2) = 120
- The word consists of two A's and two R's = 4!/(2!2!) = 6
- The word consists of two A's and two different letters = (4C2)(4!/2!) = 72
- The word consists of two R's and two different letters = (4C2)(4!/2!) = 72

TOTAL = 120 + 6 + 72 + 72 = 270

Answer: D

CrackverbalGMAT · Answer

Concepts:   (a) The number of arrangements of n things all taken together = n!

(b) Out of n things, x things of 1 type are similar, y things of another type are similar and so on, then the total arrangements =  \frac{n!}{x!y!...}

ARRANGE has 2 A's, 2 R's and 1 each of N, G and E (There are 5 distinct letters)

(i) all 4 letters are different

Choices = Choose 4 letters out of 5 = 5C4 = 5

Arrangements = 4! = 24 words

Total words of this type = 5 * 24 = 120

(ii) 2 similar and 2 different: for eg we choose A, then the 2 remaining letters can be chosen from R, N, G or E

Choices = choose 1 of A or R and then choose 3 among 4 remaining = 2C1 * 4C2 = 2 * 6 = 12

Number of arrangements =  \frac{4!}{2!} = 12

Total words of this form = 12 * 12 = 144

(iii) 2 A's and 2 R's =  \frac{4!}{2!2!} = 6

Total number of words = 120 + 144 + 6 = 270

Option D

Arun Kumar

yashikaaggarwal · Answer

Way Time Consuming

Case 1: the word is formed using 2R and 2A (AARR can be written in)= 4!/2!*2! = 6 Ways Case 2: All four letters are different then the number of words (A,R,N,G,E) = 5C4*4! =120 Case 3: 2 letters are A and the other 2 different letters (R, N, G, E), then the number of words (A A _ _ Can be written in)= 4C2*4!/2! = 72 Case 4: Similarly with R, when 2 letters in 4 letter words are R and the other 2 different letters (A, N, G, E), then the number of words (R R _ _ Can be written in)= 4C2*4!/2! = 72 Then the number of four-letter words that can be formed=6+120+72+72=270 Answer is D

TestPrepUnlimited · Answer

We want to write the letter as AA, RR, N, G, E to begin. We have 5 different letters with possible repeats.

Case 1:  No repeats.

We have only 5 letters, pick 4 out of 5 then randomize their order.  5C4 * 4! = 5*4! = 120

Case 2:  One repeat only. (AANR, RRGE etc)

Let's say we're working with two A, then we need to pick 2 more letters out of the remaining 4 to assemble a word. That is 4C2 cases. Next, we can scramble the letters so multiply by 4!, but divide by 2! at the end to account for A repeating.

4C2 * 4! / 2! = 6 * 4* 3 = 72 . Multiply this by two to get 144 as we can use R instead of A.

Case 3:  Two repeats (AARR, ARAR).

If we scramble we have 4! cases but divide by 2! for A and R each.

4!/(2!*2!) = 4*3/2 = 6 .

In total we have 120 + 144 + 6 = 270 cases.

Ans: D

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How many four lettered words can be formed using the letters of the wo

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How many four lettered words can be formed using the letters of the wo

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