Let $a, b, c, d$ denote distinct positive odd numbers. What is the smallest value of

$$
2 a b c d-(a b c+a b d+a c d+b c d)
$$
$$
2 a b c d-(a b c+a b d+a c d+b c d)
$$

I. Solution. Transform (1) as follows:

$$
2 a b c d-(a b c+a c d+a b d+b c d)=a b c d\left[2-\left(\frac{1}{d}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a}\right)\right]
$$

The value of the first factor will be the smallest when the numbers 1, 3, 5, 7 are substituted for the variables in some order. We claim that with this selection of variables, the value of the factor in square brackets will also be the smallest. Indeed, the sum of the reciprocals of positive numbers $a, b, c, d$ is positive, and subtracting this from 2, we get the smallest possible value when the sum of the reciprocals is the largest, i.e., when the values of $a, b, c, d$ are the smallest odd numbers.

This proves the statement.

II. Solution. Substituting the smallest positive numbers, 1, 3, 5, 7, into the variables in (1), the result is

$$
2 \cdot 105-(15+21+35+105)=34>0
$$

Starting from any set of four numbers that satisfy the condition, by always decreasing the numbers by two, we can eventually reach a sequence of different odd numbers, where the smallest number becomes 1, the next 3, the next 5, and the last 7.

It is therefore sufficient to show that decreasing any of the variables will also decrease the value of the expression. Due to the symmetry of (1), it does not matter which variable we decrease. Let's replace $a$ with the next smaller odd number:

$$
2(a-2) b c d-[(a-2) b c+(a-2) b d+(a-2) c d+b c d]
$$

The change is:

$$
N=4 b c d-2 b c-2 b d-2 c d
$$

We need to show that this is positive. Suppose that among the three numbers, $b$ is the largest and $d$ is the smallest: $b>c>d$. Then:

$$
\frac{1}{2} N=b(2 c d-c-d)-c d>c(2 c d-c-d)-c d
$$

It is therefore sufficient to show that

$$
2 c d-c-d>d
$$

or

$$
2 d(c-1)>c
$$

which is indeed true, since $d \geq 1$ implies $2 d(c-1) \geq 2 c-2=c+(c-2)>c$, because $c-2 \geq d \geq 1$.

In other words, (1) will be the smallest when the values of the variables are the smallest, i.e., 1, 3, 5, and 7.