In how many different ways can two pennies, three nickels, four dimes, and five quarters be arranged in a row?
if I had $2 five pennies and three quarters how much money do I have
To solve this problem, we need to use the concept of permutations.
Permutations are arrangements of objects where the order matters. In this case, we need to arrange two pennies, three nickels, four dimes, and five quarters in a row.
First, let's calculate the number of ways to arrange the coins when there are no restrictions. We can accomplish this by finding the factorial of the total number of coins.
The total number of coins is 2 + 3 + 4 + 5 = 14.
The factorial of 14, denoted as 14!, is the product of all positive integers from 1 to 14:
14! = 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Now, we need to consider that some coins are identical to each other. To account for this, we divide the total permutations by the factorials of the identical groups.
In this case, we have three groups of identical coins: two pennies, three nickels, and four dimes.
The factorial of two, three, and four are 2!, 3!, and 4!, respectively.
Now, let's calculate the number of arrangements:
Number of arrangements = 14! / (2! * 3! * 4!)
Calculating this expression will give us the answer to the number of different ways the coins can be arranged in a row.
Assuming that coins of equal vale cannot be distinguished from each other,
(2+3+4+5)! / 2!3!4!5! = 2522520