Question

The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the numbers is $10$ and the sums of the numbers placed diagonally are equal is:

• A. $4$
• B. $8$
• C. $24$
• D. $6$

Answer 1

Answer: (B)

$8$

According to question $x+b=a+y$ and $x+y+a+b=10$

So $x+b=5$ and $a+y=5$

Given: all number should be different and two pairs whose sum is $5$.

We have $2$ pair of natural number that is $(1,4),(2,3)$ whose sum is $5$.

We can choose $1$ pair among $2$ of them for $(x,b)$ by $2!$ ways and they can rearranged in $2!$ ways.

So total ways for $(x,b)$ will be $2!\times 2!$

Remaining one pair will be for $(a,y)$ and they can be arranged by $2!$ ways.

So total ways for $(y,a)$ will be $2!.$

Combining both situation together we get $2!\times 2!\times 2!=8$