If all the words that can be formed from the letters of the word GARNE

Question

If all the words that can be formed from the letters of the word GARNE are taken and arranged in alphabetical order, then the rank of the word RANGE is?
A.101
B.100
C.102
D.78
E.99

Parvezahmed2210 · Answer

Given word: GRANE. 
Alphabetical order: A,E,G,N,R
Keeping A as the first alphabet, total number of combinations: A*4*3*2=24
Similarly, for rest of the 3 alphabets: E, G, & N = 24*3 = 72
Total combination possible with A,E,G,N as the first alphabet: 24+72 =96

Now,Now to identify the rank of RANGE= 
R as the first alphabet & A is the second alphabet, total number of worlds possible= R*A*3*2*1 = 6

Therefore: Range should come as =96+6 = 102 word. (since N is the last alphabet, when R is locked as first alphabet of the word).

I am not sure, if there is any simple way to get this math. please suggest.

IanStewart · Answer

The five letters are different, so there are 5! = 120 'words' we can make in total. Of these, 1/5 will start with R, so 24 words start with R. That's the last letter alphabetically, so the last 24 words in our list begin with R, or the 97th through 120th ranked words.

Of those 24 words, exactly 1/4 of them, or 6 of them, will have an 'A' in the second position. A is the first letter alphabetically, so these six words will come first in our alphabetical ranking of the words beginning with R. So they'll occupy the 97th through 102nd positions in our list. Finally, of the six words you can make with E, G, N, the word "NGE" is last in alphabetical order, so RANGE must be the 102nd word overall.

CrackverbalGMAT · Answer

By the rule of arrangements, the number of arrangements of n distinct letters all taken together (no repetition) = n!

Since we are arranging the letters in Alphabetical order:

With A in the first place and with the remaining 4 letters the number of words formed = 1 * 4! = 24

Similarly with E, G, N we can form 24 words each.

Total words formed with A, E, G and N in the 1st place = 24 * 4 = 96.

The 97th word starts with R

RA are in order as per the alphabets

With the remaining 3 alphabets N, G and , the number of words formed = 1 * 1 * 3! = 6

These are RAEGN, RAENG, RAGEN, RAGNE, RANEG, RANGE

Therefore the rank of the word RANGE is 96 + 6 = 102

Option C

in this case, listing of the words was a feasible Option.

But if we had more letters, then a possible method is to keep counting until we have 2 letters remaining.

We know that with R and A as the 1st 2 letters in alphabetical order, we get 1 * 1 * 3! = 6 words

With R, A and E we get 2 words, R, A and G we get 2 words and with R, A and N, we get 2 words

With R, A and N the two words arranged alphabetically would be RANEG and RANGE.

Therefore RANGE is the last of the 6 words and therefore its rank is 96 + 6 = 102

Arun Kumar

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If all the words that can be formed from the letters of the word GARNE

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If all the words that can be formed from the letters of the word GARNE

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