(i) There are \\\\(6\\\\) different letters in the word MONDAY.Number of \\\\(4\\\\)-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of \\\\(6\\\\) different objects taken \\\\(4\\\\) at a time, which is \\\\(^6 P_4\\\\).Thus, required number of words that can be formed using \\\\(4\\\\) letters at a time is\\\\(\\displaystyle ^6P_4=\\frac {6!}{(6-4)!}=\\frac {6!}{2!}=\\frac {6\ imes 5\ imes 4 \ imes 3\ imes 2! }{2!}=6\ imes 5\ imes 4\ imes 3=360\\\\)(ii) There are \\\\(6\\\\) different letters in the word MONDAY. The first place can be filled in \\\\(6\\\\) ways.Second place can be filled by any one of the remaining \\\\(5\\\\) letters. So, second place can be filled in \\\\(5\\\\) waysThird place can be filled by any one of the remaining \\\\(4\\\\) letters. So, third place can be filled in \\\\(4\\\\) waysSo, on continuing, number of ways of filling fourth place in \\\\(3\\\\) ways , fifth place in \\\\(2\\\\) ways, six place in 1 way.Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is \\\\( 6 \ imes 5\ imes 4 \ imes 3\ imes 2 \ imes 1 =6!\\\\)Alternative Method:Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutation of \\\\(6\\\\) different objects taken \\\\(6\\\\) at a time, which is \\\\(^6 P_6=6!\\\\)Thus, required number of words that can be formed when all letters are used at a time= \\\\(6!= 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1=720\\\\)(iii) Total number of letters in the word MONDAY is \\\\(6\\\\).Number of vowels are \\\\(2 (O, A )\\\\)Six letters word is to be formed.\\\\(\\Box \\Box \\Box\\Box \\Box \\Box\\\\)First letter should be a vowel. So, the rightmost place of the words formed can be filled only in \\\\(2\\\\) ways.Since the letters cannot be repeated , the second place can be filled by the remaining \\\\(5\\\\) letters. So, second place can be done in \\\\(5\\\\) waysSimilarly, third place in \\\\(4\\\\) ways , fourth place in \\\\(3\\\\) ways, fifth place in \\\\(2\\\\) ways, sixth place in 1 way.Hence, required number of words that can be formed using four letters of the given word is \\\\(2\ imes 5\ imes 4\ imes 3 \ imes 2 \ imes 1 =240\\\\)