Solution: 2018 Winter Midterm - 1

Author: Michiel Smid

Question

Consider strings consisting of 12 characters, where each character is an element of the set $\{a,b,c,d,e\}$. The positions in such strings are numbered as $1,2,3,\dots,12$.
How many such strings have the property that

each even position contains an element of $\{a,b,c\}$, and
each odd position contains an element of $\{d,e\}$?

(a)

$6^3 \cdot 6^2$

(b)

$5^{12}$

(c)

$6^6$

(d)

None of the above.

Solution

Let’s break this down into 2 cases:

A = Each even position has 3 choices: $ a, b, c $

There are 6 even positions, so there are $ 3^6 $ ways to choose the characters for the even positions.

$ |A| = 3^6 $

B = Each odd position has 2 choices: $ d, e $

There are 6 odd positions, so there are $ 2^6 $ ways to choose the characters for the odd positions.

$ |B| = 2^6 $

$ |A \cap B| = 3^6 \cdot 2^6 $

$ |A \cap B| = 3^{12} $

$ |A \cap B| = 6^6 $