Question

How many sequences of zeros and/or ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both?

$190$

$192$

$211$

$380$

$382$

Answer: E

Concepts:inclusion-exclusion combinations

Difficulty rating: 1660

Solution:

Let $A$ be the sequences in which all zeros are consecutive and $B$ those in which all ones are consecutive.

For $A,$ there is one all-ones sequence, $20$ sequences with exactly one zero, and $\binom{20}{2}=190$ sequences with two or more zeros (choose the first and last zero position). So $|A|=1+20+190=211,$ and by symmetry $|B|=211.$

A sequence in $A\cap B$ is a block of zeros followed by a block of ones, or the reverse; there are $|A\cap B|=40$ of these.

Therefore $|A\cup B|=211+211-40$ $=382.$

Thus, the correct answer is E.

2012 AMC 12B Problem 12

Answer: E

Difficulty rating: 1660

Solution:

Problem 12 in Other Years