What is the tens digit in the sum 7! + 8! + 9! + … + 2006! ?

2006 AMC 10B Problem 11

Attempt Problem 11 of the 2006 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 10B solutions, or check the answer key.

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11.

What is the tens digit in the sum

7! + 8! + 9! + \cdots + 2006!\,?

1

3

4

6

9

For

n\ge 10,

n!

ends in at least two zeros

Only

7!+8!+9!

affects the last two digits

For

n\ge 10,

n!

is divisible by

100,

so it does not affect the last two digits.

The tens digit comes from

7!+8!+9!

=5040+40320+362880

=408240,

whose tens digit is

4.

Thus, the correct answer is C.