Permutation

3. In how many ways can the letters of the word ‘DAUGHTER’ be arranged so that the vowels are always separated?

Ans: Arrangement of 3 vowels coming together=4320

Again the total arrangements of 8 different letters taking all together, without any restriction 8! ways =8.7.6.5.4.3.2.1=40320

So the required arrangement=40320-4320=36,000 

4. In how many ways can the letters of the word ‘JALPAIGURI’ be arranged?

Ans: There are 10 letters. Here A is repeated 2 times, I for 2 times.

So the required number of arrangement=10!/(2!2!)=9,07,200

5. In how many ways can the letters of the word ‘BALLOON’ be arranged, so that two L’s do not come together?

Ans: There are 7 letters in the word BALLOON, of which L is repeated 2 times, O is repeated 2 times. So the no. of arrangement without restriction=

7!/(2!2!) =1260.

Now consider  2 L’s as one letter, so now we have only 6 (=1+5) letters, arrangement of which can be done 6!/2!=360 ways.

So the required arrangement, in which 2 L’s do not come together=1260-360=900

6. How many 6 – digits numbers can be formed using the digits 2,0,1,3,2,3?

Ans: There are 6 digits in which 2 occur 2 times and 3 also occur 2 times.

  So the no. of arrangement of 6 digits =6!/(2!2!)=180.

In theses 180 no. there are few no. starting with 0, so in these cases numbers will be 5 digits.

No. of such 5 digits numbers=5!/(2!2!)=30

So the required no. of 6 digit no.=180-30=150

7. In how many ways 8 boys can form a ring?

Ans: The required no. of ways=7!=5050

8. 6 persons sitting in chairs in a circle attend a committee. In how many can they arrange themselves?

Ans: Possible number of arrangement=(6-1)!=5!=120

9. In how many ways can 5 boys and 5 girls can take their seats in a round table, so that no two girls will sit side by side?

Ans: If one boy takes his seat anywhere in a round table, then remaining 4 boys can take seats in 4!=24 ways. In each of these 24 ways, between 5 boys, if 5 girls take their seats then no two girls will be side by side. So in this way 5 girls may be placed in 5 places in 5! Ways.

Again the first boy while taking seat, may take any one of the 10 seats, i,e, he may take his seat in 10 ways.

So the required no. of ways=24 x 5! X 10=24 x 120 x 10=28800