Aptitude - Permutation and Combination - Discussion

10C4 *4!
10C4 = (10*9*8*7) / (4*3*2*1) =(10*3*7) = 210,
4! = 4 * 3 * 2 * 1 = 24.

So Obviously, 10C4 * 4! = 5040.

We use permutation for selection and arranging.
They asked two processes in question.

step 1: Selection- to select 4 letters from 10 letters,
step 2: Arranging- to arrange 4 letters in different ways.

Why use arranging?

Because, by arranging the 4 words in different ways we can get many words, which can be meaningful or non-meaningful.
So, for permutation = n!/(n-1)!

We select 4 letters from LOGARITHMS in 10C2 then we arrange the 4 letters in four factorials.

¹⁰C₄ means 10x 9 x 8 x 7/ (1 x 2 x 3 x 4).

¹⁰P₄ means 10 x 9 x 8 x 7.

As we are considering order we hav to use permutations.

@Shashwat.

When we select the things and then arrange them then we use permutations.

But when we only have to select then we use combinations.

In the above question we are selecting and arranging them too that's the reason we use permutations.

If repetition is allowed then the answer is 10^4=10000!.
The no of possible letters to occupy each of the four spaces is 10.
_ _ _ _
10*10*10*10.

If repetition is not allowed,
10*9*8*7.

nCr = ( 1 / r! ) * nPr.

nPr = r! * nCr.

That's why here nPr (permutation) is used.

if repetition allowed then how to solve it? Please tell me.

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.

Its very easy, no need to get confused with permutation.

Divide the statement in two part.

From given 10 character, we have to choose 4 of them only.
So, 10C4 = 10*9*8*7/4! = 210.

Now we have to arrange there 4 character.
So, it would be simply 4! because there is no repetition of word.

Total - 210*24 = 5040.

Moral - choose 4 word out of 10 then arrange them by 4!.