## 24. Permutations

A certain three-digit number will increase by 45 if the two rightmost digits are swapped, and it will decrease by 270 if the two leftmost digits are swapped. What will happen to this number if the two outermost digits are swapped?
24. Let $c, d$ and $u$ be the number of hundreds, tens, and units of a given number, respectively. According to the problem, we have the following equations:

$$
\begin{aligned}
& (100 c+10 d+u)=(100 c+10 u+d)-45 \\
& (100 c+10 d+u)=(100 d+10 c+u)+270
\end{aligned}
$$

From this, we get

$$
\begin{gathered}
9 d-9 u+45=0 \\
90 c-90 d-270=0
\end{gathered}
$$

That is,

$$
u=d+5, \quad c=d+3
$$

If we swap the two extreme digits, the difference between the resulting values will be

$$
\begin{gathered}
(100 u+10 d+c)-(100 c+10 d+u)= \\
=99(u-c)=99[(d+5)-(d+3)]=198
\end{gathered}
$$

Thus, with this permutation of digits, the number will increase by 198.