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Permutations with Identical Items. Created by: K Wannan Edited by: K Stewart. JENN JOHN. Jenn Jnen Jnne Ejnn Enjn Ennj Njen Njne Nejn Nenj Nnje Nnej 12. John Jhon Jonh Jhno Jnoh Ojhn Ojnh Ohnj Onjh Ohjn Etc….. 24. Why did Jenn have less permutations than John?.
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Permutations with Identical Items Created by: K Wannan Edited by: K Stewart
JENN JOHN Jenn Jnen Jnne Ejnn Enjn Ennj Njen Njne Nejn Nenj Nnje Nnej 12 • John • Jhon • Jonh • Jhno • Jnoh • Ojhn • Ojnh • Ohnj • Onjh • Ohjn • Etc….. • 24
Why did Jennhave less permutations than John? • The double letter reduces the number of permutations • The two n’s can trade places and it be the same permutation as another.
Permutations with Identical Items • To find the number of permutations with identical items, divide the total number of arrangements by the number of ways to arrange the identical items only. How many ways can you arrange n1.n2 ? How many ways could you arrange the four letters of J.e.n1.n2? 2! 4! 4!/2! = 24/2 =12
Example • How many permutations are there of the letters in the name Kristina
Permutations of Identical Items • The number of permutations of a set of n items with a, identical items is:
Example • Tiles used to surround a bathroom floor. There are 4 yellow, a blue, green, red and grey tile. How many patterns can be made? There are 8 tiles in total, with 4 of them being identical
When Some Items Are Alike There will be times when some of the items in the set are alike. Suppose we are asked to find the number of ways of arranging the letters in the word EXCELLENT. For example; So far we know that factorials will be used… The existence of duplicate items changes this slightly. a simply represents the # of times the first duplicate appears. b simply represents the # of times the second duplicate appears. And so on…
Suppose we are asked to find the number of ways of arranging the letters in the word EXCELLENT. In this case we have;3 E’s (a = 3)2 L’s (b = 2) Meaning that there are 30240 different ways to arrange the letters in the word EXCELLENT.
Example • How many permutations are there of the letters of WANNAN? We have more than one identical item. In this case we have 3 n’s and 2 a’s You multiply all repeated items together before dividing (or use brackets around denominator).
Permutations of Repeated Identical Items • The number of permutations of a set of n items with a, b, c, which are different identical items is:
Examples a) How many Permutations are possible of the letters of the word bookkeeper? b) Barb has three different items of clothes to display in her store window; five sweaters, three t-shirts, and four pairs of pants. How many ways can she arrange them?
Homework • Page 245 #1-5, 7-12, 17a