2018 AMC 10A Problem 3

A unit of blood expires after

10! = 10 \cdot 9 \cdot 8 \cdots 1

seconds. Yasin donates a unit of blood at noon on January

1.

On what day does his unit of blood expire?

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Solution

We can divide

10!

60,

60,

and

24

to get the number of days that it takes for a unit of blood to expire. The first division cancels a

6

and

10.

The second division cancels

3, 4,

and

5.

The final division cancels

8

and turns the

9

into a

3.

This leaves

2, 7,

and a

3,

which multiply to

42.

There are

31

days in January, so by February

1,

the blood only has

42 - 31 = 11

days left.

11

days from February

1

would make the blood expire on February

12.

Thus, E is the correct answer.