2018 AMC 10A Problem 3

Question

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?

January

2

January

12

January

22

February

11

February

12

Convert

10!

seconds into days

Use

60\cdot60\cdot24

seconds per day

We can divide

10!

by

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.

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