Aptitude - Permutation and Combination - Discussion

Formula for Permutation nPr= n!/(n-r!)
For 10P4= 10!/(10-4!).
=> 10!/6!.
=>10x9x8x7x6x5x4x3x2x1
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6x5x4x3x2x1
=>6x5x4x3x2x1 in both numerator and denominator will get cancelled.

Hence 10x9x8x7=5040.

The given word consists of 10 letters.

No letters are repeated
we have to form a four-letter word and letters should not be repeated so;
_ _ _ _
Consider 4th place we have now 10 letters and one place to fill, So 10C1 possibilities are there
Now consider 3rd place we have 9 letters(one already used)and one place to fill, so 9C1 possibilities are there.

Now consider 2nd place we have 8 letters and one place to fill, so 8C1 possibilities are there
similarly, for 1st place, 7C1 possibilities are there.

Now multiplying them =10C1x9C1x8C1x7C1=10x9x8x7=5040.

Thanks for explaining the solution.

How 4 comes? Please explain.

There is not repetition getting, Logarithm all the words are different.

So permutations are taken.

Am I right?

If repetitions is allowed the formula would be 10^4.

What is the answer if repetition is allowed?

If the letters can be repeated then the answer would be 10^4.

Because if the word is _ _ _ _.

The 1st blank can be filled in 10 ways.

The 2nd blank can be filled in 10 ways again (since letters can be repeated).

Similarly, 3rd and 4th blanks can also be filled in 10 ways.

So, according to the fundamental principle the number of ways =10*10*10*10=10^4.

Firstly we have to check if any alphabet is repeated in the word "LOGARITHMS".

No, alphabets is repeated so it can be treated as any other general case. We are using permutation because it is not just the selection of 4 alphabets from the given word but the rearrangement of these alphabets in order to form a word (given: The words may or may not have any meaning). Now there are 10 alphabets & we must find out the number of 4 lettered words that can be formed which can be done in 10p4 ways. We need not multiply with 4! because have already considered the rearrangements (i.e usage of the concept of permutations).

I agree @Ankur.

It is first we are selecting 4 letter from 10 letters.

i.e. C (10, 4) = 210.

Then arranging 4 letter in different ways. i.e. 4!= 24.

Therefore, it is selecting and arranging (combination of permutation and combination) =210 * 24 = 5040.