4.3 Permutations With Some Identical Items Permutations With Some Identical Items When dealing with permutations of items that include identical items of one type, identical items of another type, and so on, you can use the formula Example 1 Permutations With Some Identical Elements Compare the different permutations for the words DOLE, DOLL, and LOLL. Solution For DOLE are 24 permutations of the four letters in DOLE. For DOLL Only 12 are actually different order, thus the number of different arrangement is 12. For LOLL Thus, the number of permutations is 4. You can generalize the argument in Example 1 to show that the number of permutations of a set of items of which are identical is . Example 2 Tile Patterns Tanisha is laying out for the edge of a mosaic. How many patterns can she make if she uses four yellow tiles and one each of blue, green, red, and grey tiles? Solution and Tanisha can make 1680 different patterns with the eight tiles. Example 3 Permutation With Several Sets of Identical Elements The word is unusual in that it has three consecutive double letters. How many permutations are there of the letters in ? Solution There are 151 200 permutations of the letters in . The number of permutations of a set of objects containing identical objects of one kind, identical objects of a second kind, identical objects of a third kind, and so on is Example 4 Applying the Formula for Several Sets of Identical Elements
4.3 Permutations With Some Identical Items Permutations With Some Identical Items When dealing with permutations of items that include identical items of one type, identical items of another type, and so on, you can use the formula Example 1 Permutations With Some Identical Elements Compare the different permutations for the words DOLE, DOLL, and LOLL. Solution For DOLE are 24 permutations of the four letters in DOLE. For DOLL Only 12 are actually different order, thus the number of different arrangement is 12. For LOLL Thus, the number of permutations is 4. You can generalize the argument in Example 1 to show that the number of permutations of a set of items of which are identical is . Example 2 Tile Patterns Tanisha is laying out for the edge of a mosaic. How many patterns can she make if she uses four yellow tiles and one each of blue, green, red, and grey tiles? Solution and Tanisha can make 1680 different patterns with the eight tiles. Example 3 Permutation With Several Sets of Identical Elements The word is unusual in that it has three consecutive double letters. How many permutations are there of the letters in ? Solution There are 151 200 permutations of the letters in . The number of permutations of a set of objects containing identical objects of one kind, identical objects of a second kind, identical objects of a third kind, and so on is Example 4 Applying the Formula for Several Sets of Identical Elements