No. of letters in INDEPENDENCE : 12

repetitions : 4 Es, 2 Ds, 3 Ns.

therefore no. of ways of arranging the letters of the word INDEPENDENCE is12! / (4!*3!*2!)since there are twelve letters in total and 3 letters are repeated 4 times,3 times and 2 times respectively.

1)WORDS START WITH P

P is fixed for first place. remaining no. of letters is 11 and here also there are 4 Es, 2 Ds, 3 Ns.

therefore , no. of ways of arranging is11! / (4!*3!*2!).

2)ALL VOWELS OCCUR TOGETHER

vowels present in the INDEPENDENCE are i and e. But e occurs 4 times. so, totally we must consider 5 vowels.remaining no. of letters is 7 whre there are 3 Ns and 2 Ds.

as the 5 vowels are always together , we must consider them as one object. therefore , there are 8 objects totally (7 letters and one set of vowels).

these 8 objects can be arranged in8! / (3!*2!) * 5! / 4! wayssince, in 8 objects , n and d are repeated 3 times and 2 times respectively and we multiply by(5! / 4!) since the vowels can be arranged internally in that many ways.