A magician wants to hide four identical balls in 16 boxes. More than one ball can be placed in a box. In how many ways can the balls be hidden?
When solving any problem involving combinations, we firstly need to identify whether we’re dealing with counting with or without replacement and whether or not order matters. Once we have done this, we can apply the appropriate formula. Since more than one ball can be placed in a box, we are counting with replacement. This means that our problem will include repetition.
In this question, it is also clear that the order does not matter. When dealing with a problem with repetition and where order does not matter, we can use the formula 𝑛 plus 𝑟 minus one factorial divided by 𝑟 factorial multiplied by 𝑛 minus one factorial, where 𝑛 is the total number and 𝑟 is the number being chosen. In this question, the magician has 16 boxes. Therefore, 𝑛 is equal to 16. He wants to hide four identical balls. Therefore, 𝑟 is equal to four.
Our numerator is 16 plus four minus one factorial. Our denominator is four factorial multiplied by 16 minus one factorial. This simplifies to 19 factorial divided by four factorial multiplied by 15 factorial. The factorial of a number is the product of that integer and all smaller positive integers. For example, four factorial is equal to the product of four, three, two, and one. Four multiplied by three is equal to 12. Multiplying this by two gives us 24. And multiplying this by one gives us 24. Therefore, four factorial is equal to 24.
Using this method to calculate 15 factorial and 19 factorial would be very time consuming. As a result, we can use the factorial button on the calculator denoted by 𝑥 and then an exclamation mark. Typing 19 factorial divided by four factorial multiplied by 15 factorial into our calculator gives us 3876. The total number of ways that the magician can hide the balls is 3876.