# Problem of the Day: 3/1/13

How many palindromic 10-digit phone numbers can exist if the first three digits must be different?

Solution to yesterday’s problem:

Let’s write two rates to represent how much water the tap can fill in one minute and how much the drain can empty in one minute.

The tap can fill the tub in 30 minutes, so it can fill 1/30 of the tub in one minute. The drain can empty the tub in 45 minutes, so it can drain 1/45 of the tub in one minute. We can write the amount that the tub fills in one minute as

$\frac{1}{30}-\frac{1}{45}$

Simplifying this fraction,

$\frac{3}{90}-\frac{2}{90}$

So the tub will fill by $\frac{1}{90}$ in one minute. Logically, it will take 90 minutes to fill the entire tub.

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