Three people, A, B and C, stand in a row. How many different orders are there?
Once the first place is taken, two people are left, and those two can stand in only two orders, which fixes the last place. That gives 3 × 2 × 1 = 6. The tree shows the choices shrinking at every step, right in the number of branches.
Two of the three cards 1, 2 and 3 are placed side by side to make a two-digit number. How many numbers can be made?
The card used in the tens place cannot be used again in the ones place. So the tens place has 3 choices and the ones place has 2, giving 3 × 2 = 6. Nothing may be used twice, and the branches thin out because of it.
Two players, A and B, play one round of rock-paper-scissors. How many ways can the two hands come out?
Whatever A throws, B still picks from rock, paper and scissors. Nothing gets used up, so the count is 3 × 3 = 9. Three of them are draws, and the other six decide a winner.
A coin is tossed three times in a row. How many sequences of heads and tails are there?
Whatever came before, the next toss is still heads or tails. Nothing gets used up, so the count is 2 × 2 × 2 = 8. Every node splits the same way at every level, and that is what separates this from an arrangement.
Two of the four colors red, blue, yellow and green are chosen and placed left and right to make a flag. How many arrangements are there?
The left half has 4 choices and the right half has 3, since the color already used is gone, giving 4 × 3 = 12. Red with blue and blue with red count as different flags, because the two halves sit in fixed places.
Choose one main dish from rice, bread and noodles, and one drink from tea and milk. How many combinations are there?
The main dish splits into three, and each of those splits into two drinks. The two levels draw from different sets, so nothing taken above is gone below. That gives 3 × 2 = 6. This is the shape where plain multiplication is enough.
Two people are chosen from A, B, C and D. How many ways are there?
A with B and B with A are the same pair, so each pair is counted once. The second name is taken only from the ones after the first, and D has no one after it, so no branch grows from it. That leaves 6, exactly half of the 12 you get when the order matters.