G3.4 When choosing, without replacement, 4 out of 30 labelled balls that are marked from 1 to 30 , there are $r$ combinations. Find the value of $r$.
$$
r={ }_{30} C_{4}=\frac{30 \times 29 \times 28 \times 27}{1 \times 2 \times 3 \times 4}=27405
$$

Remark: If the question is changed to: Choose 4 out of 30 labelled balls that are marked from 1 to 30 with repetition is allowed, there are $r$ combinations. Find the value of $r$.
We shall divide into 5 different cases:
Case 1: All 4 balls are the same number, 30 combinations
Case 2: XXXY, where X, Y are different numbers, ${ }_{30} C_{2}=435$ combinations
Case 3 XXYY, where $\mathrm{X}$, Y are different numbers, ${ }_{30} C_{2}=435$ combinations
Case $4 X X Y Z$, where $X, Y, Z$ are different numbers, $30 \times 29 \times 28 \div 2=12180$ combinations
Case 5 XYZW, where X, Y, Z, W are different numbers, ${ }_{30} C_{4}=27405$ combinations
Total number of combinations $=30+435+435+12180+27405=40485$