2019 AMC 12B Problem 23

Attempt Problem 23 of the 2019 AMC 12B 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 2019 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

How many sequences of

0

s and

1

s of length

19

are there that begin with a

0,

end with a

0,

contain no two consecutive

0

s, and contain no three consecutive

1

55

60

65

70

75

Between consecutive

0

s there is a block of one or two

1

With

k

zeros there are

k-1

blocks of size

1

2

whose sizes sum to

19-k

No two

0

s are adjacent, so the

0

s are separated by blocks of

1

s, each of size

1

2

(never

3

). If there are

k

zeros, there are

k-1

such blocks summing to

19-k

ones.

The number of size-

2

blocks is

(19-k)-(k-1)=20-2k,

which must satisfy

0\le20-2k\le k-1,

i.e.

7\le k\le10.

Summing

\binom{k-1}{20-2k}

over

k=7,8,9,10

gives

\binom66+\binom74+\binom82+\binom90

=1+35+28+1

=65.

Thus, C is the correct answer.

Problem 23 in other years