In this lesson on Permutations, will learn more concepts on permutations and solve problems on them.

Without allowing things to repeat,

Without allowing any letter to repeat, 4 different letters can be arranged in 24 ways taking all at a time.

Now the next formula:

Definition of

Without allowing any letter to repeat,

Without allowing any letter to repeat, 12 permutations can be formed taking any 2 out of 4 different things.

Consider four different letters

Without allowing any letter to repeat, taking any 2 out of 4 different letters a, b, c, d, listed below are the

In

In

Solved Examples:

1. Find n if

Solution:

So n = 6

2. Find r if

Solution:

Therefore,

Definition of

Useful Tip:

Without allowing things to repeat, listed below are the:

Allowing each thing to repeat (only 2 times here), listed below are more:

In how many ways 4 letters a, b, c, d can be arranged taking 3 at a time in each permutation?

Answer:

Include repetitions where there is scope and no restriction.

(Letters can repeat as there is scope and here repetitions are not prevented)

Using formula

You will see three types of permutations in the 64 arrangements.

Solved ExamplesType 1: When a letter repeats all three times. Eg:aaa

Type 2: When a letter repeats two times. Eg:aab

Type 3: When no letter repeats. Eg:abc

1. How many ways 5 prizes can be given away to 4 students, if each student can receive all the prizes?

Solution:Prizes are given away to students.

1

Each of 2

5 prizes therefore can be given away in

2. How many numbers having 4 digits can be formed with non-zero even digits?

Solution:Non-zero even digits are: 2, 4, 6 and 8

Digits repeat in numbers and unless restricted, include repetitions.

Use n

3. How many words having 5 distinct letters can be formed with the letters in the word

Solution:

Unless restricted, letters normally repeat in words. [ like in

But, the word “distinct” in the question restricts letters from repeating.

Recall

Without allowing things to repeat, taking all,

Here too, without allowing letters to repeat, taking all five letters from the word “MATHS”, 5! arrangements can be made.

The 120 arrangements or permutations are 120 words.

Its importance (why different) will get clear with the formula in hand now

How many ways you can

Only 6 ways. Listed below are the

Why not more? Not possible!

Rearrange A with A and B with B in one sample permutation: AABB.

Look what happens! You get AABB. Oh, my! The same arrangement.

Got hold of why “different” was used to qualify n?

With four letters

permutations can be made

Generalize. How? As follows:

Solved Examples:

1. How many words can be formed taking all the letters in the word

Solution:

Out of the 11 letters in the word:

4 are identical of one type (the 4 Is); 4 are identical of a second type (the 4Ss); and 2 are identical of a third type (the two Ps)

So,

arrangements can be formed.

Each arrangement is a permutation and a word too.

2. Six coins are tossed at a time. How many outcomes show 4 heads and 2 tails?

Solution:

Here, 4 Heads are identical of one type and 2 Tails are identical of a second type.

Apply

, and permutations are