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A six letters word is formed using the letters of the word LOGARITHEM with or without repetition. Find the number of words that contain exactly three different letters. |
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Answer» Solution :We have letters L, O, G, A, R, I, T, H, M. Words contain exactly THREE different letters. Three letters can be selected in `.^(9)C_(3)` ways. Now we have following cases for the occurrence of these three letters. Case I : Occurrence of letters is 4,1,1 The letter which is occurring four TIMES can be selected in `.^(3)C_(1)` ways. Then letters can be arranged in `(6!)/(4!)` ways. So, number of words in this case are `.^(3)C_(1)xx(6!)/(4!)=90` Case II : Occurrence of letters is 3,2,1 The letter which is occurring three times can be selected in `.^(3)C_(1)` ways. The letter which is occurring two times can be selected in `.^(2)C_(1)` ways. Then letters can be arranged in `(6!)/(3!2!)` ways. So, number of words in this case are `.^(3)C_(1)xx .^(2)C_(1)xx(6!)/(3!2!)=360` Case III : Occurrence of letters is 2,2,2 Since each letters is occurring TWICE, number of words are `(6!)/(2!2!2!)=90` So, TOTAL number of words `=.^(9)C_(3)xx(90+360+90)` `=84xx540` =45360 |
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