Answer to Question #148409 in Statistics and Probability for Bozzy

Question

Suppose you have 9 different (distinguishable) coins, and 4 different (distinguishable) juke- box slots.

1. How many ways can you insert the 9 coins into the jukebox slots, if the order in which the coins are inserted does not matter?
2. How many ways can you insert the 9 coins into the jukebox slots, if the order in which the coins are inserted into each jukebox does matter?
3. How many ways can you insert 6 of the coins into one of the jukebox slots, if the order in which the coins are inserted matters?

Answer 1

1) Let's take the ordered sequence of coins and put them into the boxes one by one.

Number of possible boxes for 1 coin = 4.

Then, the number of ways you can insert 9 coins into the jukebox slots, if the order in which the coins are inserted does not matter = (number of boxes)^{number of coins inserted} = 4⁹ = 262144

2) To calculate the number of ways, you can insert the 9 coins into the jukebox slots, if the order in which the coins are inserted into each jukebox does matter (N), we need to multiply the number of possible sequences of coins (9!) by the number of possible splittings of these sequences into 4 parts (10*1+9*2+8*3+7*4+6*5+5*6+4*7+3*8+2*9+1*10)

N = 9!*(10*1+9*2+8*3+7*4+6*5+5*6+4*7+3*8+2*9+1*10) =

9!*(10*1+9*2+8*3+7*4+6*5+5*6+4*7+3*8+2*9+1*10) =

9!*220 = 79833600

3) Number of ways you can insert 6 of the coins into one of the jukebox slots, if the order in which the coins are inserted matters = (number of coins inserted)! = 6! = 720 (permutations without repetition)