2020 AMC 10A Problem 18

Question

Let

(a,b,c,d)

be an ordered quadruple of not necessarily distinct integers, each one of them in the set

\{0,1,2,3\}.

For how many such quadruples is it true that

a\cdot d-b\cdot c

is odd? (For example,

(0,3,1,1)

is one such quadruple, because

0\cdot 1-3\cdot 1 = -3

is odd.)

48

64

96

128

192

Only the parities of

a,b,c,d

matter

Count invertible

2\times2

matrices over the two-element field

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Only parity matters. Modulo

2

, the condition is that

ad-bc

is

1

, meaning the matrix

\begin{pmatrix}a&b\\ c&d\end{pmatrix}

is invertible over

\mathbb F_2

.

There are

(4-1)(4-2)=6

invertible

2\times2

matrices over

\mathbb F_2

. Each parity pattern lifts to

2^4=16

choices from

\{0,1,2,3\}

, so there are

6\cdot16=96

quadruples. Thus, C is the correct answer.

Problem 18 in other years