Find the number of ways in which 6 objects out of 8 can be arranged so that two particular objects are :

(a) included (b) excluded (Hint : (a) ⁶P₂ x ⁶P₄ (b) ⁶P₆)

(i) first letter is a vowel (ii) no two vowels are together

(iii) relative position of vowels and consonants remains unchanged?

 (i) How many different numbers of 6-digits can be formed with the numbers 3, 5, 7, 8, 2, 6?

(ii) How many of them are divisible by 5?

126. In how many ways can 10 students in a class be seated so that the most intelligent student and the weakest student never sit together?

(ii) no two vowels are together ?

(iii) vowels may occupy odd places ?

(i) exactly three flags can be used for a signal?

(ii) at most three flags are to be used for a signal?

(iii) at least three flags are to be used for a signal?