How many five - digit numbers can be formed with the digits 5, 4, 3, 5, 3?

In how many ways can the letters of the word ‘INTERMEDIATE’ be arranged so that:

(i) the vowels always occupy even places?

(ii) the relative orders of vowels and consonants do not change?

(i)Find the number of different words by using all the letters of the word, ‘INSTITUTION’.

In how many of them

(ii) are the three T’ s together

(iii) are the first two letters the two N’ s?

How many numbers can be formed with the digits 2, 3, 4, 5, 4, 3, 2 so that the odd digits occupy the odd places?

How many 7 - digit numbers can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?