It is 6! = 6*5*4*3*2*1 = 720
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We can clearly observe that the word "ellises" has 7 letters and three pairs of letters are getting repeated that are 'e','l' and 's'. So, Number of distinguishable permutations = 7!/(2!2!2!) = 7 x 6 x 5 x 3 = 630.
No, sometimes they will be equal (when all items being permutated are all different, eg all permutations of {1, 2, 3} are distinguishable).
First of all, find the total number of not-necessarily distinguishable permutations. There are 12 letters in hippopotamus, so use 12! (12 factorial), which is equal to 12 x 11x 10 x9 x8 x7 x6 x5 x4 x3 x2 x1. 12! = 479001600.Then count the of each letter and calculate how many permutations of each letter can be made. For example, here is 1 h, so there is 1 permutation of 1 h.H 1I 1P 60 2T 1A 1M 1U 1S 1Multiply these numbers together. 1 x1 x6 x2 x1 x1 x1 x1 x1 = 12Divide 12! by this number. 479001600 / 12 = 39,916,800 Distinguishable Permutations.
Since there are no duplicate letters in the word RAINBOW, the number of permutations of those letters is simply the number of permutations of 7 things taken 7 at a time, i.e. 7 factorial, which is 5040.
The number of permutations of the letters in PREALGEBRA is the same as the number of permutations of 10 things taken 10 at a time, which is 3,628,800. However, since the letters R, E, and A, are repeated, R=2, E=2, A=2, you must divide that by 2, and 2, and 2 (for a product of 8) to determine the number of distinctpermutations, which is 453,600.