A number has three distinct digits. The sum of the digits equals the product of the digits. How many three-digit numbers satisfy this condition?
x+y+z = x*y*z
1+2+3 = 1*2*3
...
233
To find the number of three-digit numbers that satisfy this condition, we can break down the problem into steps:
Step 1: Find all possible digits
Since the number has three distinct digits, we need to find all the possible combinations of three digits. We can start by listing all the digits from 0 to 9: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
Step 2: Find all combinations of three digits
To find all possible combinations of three digits, we can use the concept of permutations. Since the digits are distinct, we can use the permutation formula. The formula for finding the number of permutations of r elements from a set of n elements is given by nPr = n! / (n - r)!. In this case, we have 10 digits (n = 10) and we want to select 3 digits (r = 3). Therefore, the number of combinations would be: 10P3 = 10! / (10 - 3)! = 10! / 7! = (10 x 9 x 8) / (3 x 2 x 1) = 120.
Step 3: Check conditions for each combination
Now, for each combination of three digits, we need to check if the sum of the digits is equal to the product of the digits. For example, if we have the combination {1, 2, 3}, we need to check if 1 + 2 + 3 = 1 x 2 x 3. If the sum and product are equal, we count that combination as one of the satisfying numbers.
Step 4: Count the satisfying numbers
Go through all the combinations and count how many combinations satisfy the condition of having the sum of the digits equal to the product of the digits. Keep track of this count.
Once you follow these steps, you should be able to find the number of three-digit numbers that satisfy the given condition.