Indistinguishable Objects

Example: Basic Application

How many different strings can be made using all the letters from ABRACADABRA? This is very similar to the example earlier in the tutorial. Again, we identify all the unique objects: (5) A (2) B (1) C (1) D (2) R Apply what we know of the formula, n = 11, n1 = 5, n2 = 2, n3 = 1, n4 = 1, and n5 = 2. Therefore, the number of different strings is = 11! 5! 2! 1! 1! 2! = 83,160

Example: Advanced Application

Stranded on a desert island with nothing but your Discrete Mathematics textbook, you stumble across a cache of tropical fruit. You discover 3 kumkwats, 4 avacados, and 2 pugnacious pomegranates. To nourish your mind and body, you decide to figure out how many different orders you can eat the fruit. Applying the 'Indistinguishable Objects' formula, you determine that n = 9, n1 = 3, n2 = 4, and n3 = 2. So, the number of different orders to eat the fruit are = 9! 3! 4! 2! = 1260

Objects in Boxes

Example

How many ways are there to deal hands of 7 cards to each of five players from a standard deck of 52 cards? Solution: Similar to the previous example we have 52 distinguishable objects, so n = 52. Additionally, k = 6 (5 players + the undealt stack). Using the theorem as a guide, we determine the following equation: = 52! 7! 7! 7! 7! 7! 17! =69731208959821871249835089602560000