Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Parity from 2020 AMC 10A Problem - 18

What is Parity?


In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even .

Try the problem


Let $( \textbf a, \textbf b, \textbf c, \textbf d)$ 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 $ \textbf a\cdot \textbf d - \textbf b\cdot \textbf c$ is odd?

$\textbf{(A) } 48 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 128 \qquad \textbf{(E) } 192$

2020 AMC 10A Problem-18

Parity

4 out of 10

Mathematics Circle

Knowledge Graph


Parity-Knowledge Graph

Use some hints


We need exactly one term to be odd, one term to be even. Because of symmetry,let us set $\textbf a \textbf d$ to be odd and $\textbf b \textbf c$ to be even,then multiple by $2$.

Now can you complete the sum using odd and even property?

See If  $\textbf a \textbf d$ is odd, then both $\textbf a$ and $\textbf  d$ must be odd, therefore there are $2$.$2$=$4$ possibilities for $\textbf a \textbf d$.

now consider $\textbf b \textbf c$, we can say that $\textbf b \textbf c$ is even,then there are $2$.$4$=$8$ possibilities for $\textbf b \textbf c$ . However, $\textbf b$ can be odd.in that case $2$.$2$=$4$ more possibilities for $\textbf b \textbf c$. Thus there are $8$+$4$=$12$ ways for us to choose $\textbf b \textbf c$ and also $4$ ways are there to choose $\textbf a \textbf d$.

Considering symmetry, to $\textbf a \textbf d $- $\textbf b \textbf c$ be odd,there are $12$.$4$.$2$ = $96$ quadruples .So, the answer is $96$.

Subscribe to Cheenta at Youtube


Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com