Question

How many numbers can you make between 0-99 using exactly four 4s and the operations $$ +,−,×,÷?

Answer 1

To determine the number of numbers between 0-99 using four 4s and the operations +, -, ×, ÷, we can use permutations. There are a total of 24 combinations. Various examples of the combinations are provided.

Step-by-step explanation:

To find the number of numbers you can make between 0-99 using exactly four 4s and the operations +, -, ×, ÷, we need to determine all the possible combinations.

To do this, we can use the concept of permutations. Since we have 4 4s, we have 4 options for the first position, 3 options for the second position, 2 options for the third position, and 1 option for the fourth position. Therefore, the total number of combinations is 4 × 3 × 2 × 1 = 24.

Here are some examples of the numbers you can make: 4 + 4 + 4 + 4 = 16, 4 × 4 × 4 ÷ 4 = 16, 44 - 4 - 4 = 32.

Answer 2

37

Step-by-step explanation:

I find 37 numbers in that range:

{0, 1/16, 1/8, 1/5, 1/3, 1/2, 3/4, 4/5, 1, 5/4, 4/3, 2, 3, 7/2, 15/4, 4, 17/4, 9/2, 5, 6, 7, 8, 9, 12, 15, 16, 17, 20, 24, 28, 32, 36, 48, 60, 64, 68, 80}

For these numbers we have used only the indicated operations. No minus signs were used.

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Additional comment

Of the 320 ways the numbers can be combined, 15 result in division by zero. Many other results are outside the 0-99 range of interest. In total, there are 56 unique rational results.