# Permutations of objects which are not all different

- The number of permutations of n things taken all together, when p of the
things are alike of one kind, q of them alike of another kind, r of them alike
of a third kind and the remaining all different is
**n! / [p! q! r!]** - The number of permutations of n objects, of which m are of one kind and
the rest n -m of another kind, taken all at a time is
**n! / [m! (n-m)!]**

## Illustrative Examples

### Example

In how many ways can the letters of the word permutations be arranged such that

- there is no restriction
- P comes before S
- words start with P and end with S
- T's are together
- vowels are together
- order of vowels remains unchanged?

### Solution

The given word has 12 letters -two Ts and 10 different letters.

- Total number of arrangements is12!/2! = 6. 11!
- Out of above, P comes before S in half the arrangements. Hence required number of arrangements = 3. 11!
- As position of P and S is fixed, remaining 10 letters (that is, two T's and eight other different letters) can be arranged in 10!/2! = 5. 9! ways.
- Considering two T's as a block, we have to arrange 11 different things, which can be done in 11! ways.
- Considering the five vowels in given letter -E, U, A, I, O as a block, we have 8 things having 2 alike things (T's). So this can be arranged in 8!/2! = 4. 7! ways. Now within the block, 5 different vowels can be arranged in 5! ways. Hence required number of arrangements is 4. 7!5! ways.
- If order of five vowels has to remain unchanged, we can consider them like five alike things, so only one ordering is possible. Thus we have 12 things of which 2 are alike and 5 are alike. Hence required number of arrangements is 12!/[2! 5!]

## Exercise

- How many different signals can be transmitted by hoisting 3 red, 4 yellow and 2 blue flags on a pole, assuming that in transmitting a signal all nine flags are to be used?
- In how many ways can five red marbles, two white marbles and three blue balls be arranged in a row?
- In how many ways can you arrange six identical coins in a row so that you get exactly 2 heads?
- Find the number of arrangements that can be made out of the following words:

(i) BANANA

(ii) APPLE

(iii) PINEAPPLE

(iv) INDEPENDENCE

(v) ASSASSINATION - How many arrangements can be made with the letters of the word MATHEMATICS if

(i) there is no restriction

(ii) vowels occur together

(iii) all vowels don't occur together

(iv) consonants occur together

(v) M is at both extremes

(vi) order of vowels remains unchanged? - How many different numbers can be formed out of all the digits of 111223? How many of these are greater than 300000?
- How many 7 digit numbers can be formed using the digits 1, 2, 0, 2, 4, 2, 4?
- How many numbers greater than 100000 can be formed by using the digits 2, 4, 2, 3, 0, 2 taken all together?
- How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that odd digits always occupy odd places?
- If permutations of all the letters of the word AGAIN are arranged in dictionary order, which is the fiftieth word?
- In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's occur together?

## Answers

**1.**1260

**2.**2520

**3.**15

**4.**(i) 60 (ii) 60 (iii) 30240 (iv) 1663200 (v) 10810800

**5.**(i) 4989600 (ii) 120960 (iii) 4868640 (iv) 75600

(v) 90720 (vi) 415800

**6.**60, 10

**7.**360

**8.**100

**9.**18

**10.**NAAIG

**11.**151200