Are you good with money?
Probability
Level
3
In how many ways can you pay for a pair of $100 shoes using only $1, $5, $10, and $25 bills?
The answer is 242.
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1 solution
Do this case by case:
Case 1- 4 $25 bills, (1 way)
Case 2- 3 $25 bills, which leaves you with $25 you need to pay. You can use a maximum of 2 $10 bills, with $5 leftover. These $5 can be made up using a $5 bill or 5 $1 bills. Then you use $1 bills, which would leave you with $15. You will realize that there is a sequence, with 2 $10 bills, you have 2 ways. With 1 $10 bill, you have 4 ways, because the $10 bill can be replaced with 2 $5 bills or 10 $1 bills. With 0, you have 6 ways. Case 2 gives you 12 ways.
Case 3- 2 $25 bills, which leaves you with $50 you need to pay. You can use 5 $10 bills, which is 1 way. As you subtract the number of $10 bills-4,3,2,1,0, the number of ways increases by 2 each time, forming an arithmetic sequence- 1, 3, 5, 7, 9, 11. This gives you 12/2 *6 ways= 36 ways.
Case 4- 1 $25 bills, which leaves you with $75 you need to pay. You can use a maximum of 7 $10 bills, with $5 left over, which gives you 2 ways to make that $5: 1 $5 bill or 5 $1 bills. As you decrease the number of $10 bills- 7,6,5,4,3,2,1,0, the number of ways increases by 2, forming an arithmetic sequence: 2, 4, 6, 8, 10, 12, 14, 16. This gives you 18/2 *8= 72 ways.
Case 5- 0 $25 bills, which leaves you with $100 you need to pay. You can use 10 $10 bills, which gives you one way. As you decrease the number of $10 bills- 10,9,8,7,6,5,4,3,2,1,0, the number of ways increases by 2, forming an arithmetic sequence: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 This gives you 22/2 *11= 121 ways.
Add all of the cases together: 1+12+36+72+121= 242 ways.