Counting combinations

The idea

If each choice for one item can go with each choice for another, the total number of combinations is the product of the numbers of choices for the items.

Estu’s dresses and caps

Estu has 4 dresses and 3 caps. He wears one dress with one cap.

How many different dress-and-cap outfits can he make?

• 7
• 12
• 4

Make every outfit

Tap a square to make that outfit. Tap a dress or a cap to make all of its outfits at once.

Now the old safe. A lock has dials with the digits 0 to 9. Add dials and watch the number of codes.

Multiply the choices

If each choice for one item can go with each choice for another, the total number of combinations is the product of the numbers of choices for the items.

4 × 3 = 12 outfits. (For each cap there are 4 dresses: 4 + 4 + 4 = 12.)

A lock with 2 dials: 10 × 10 = 100 codes. 3 dials: 10 × 10 × 10 = 1000 codes. 5 dials: 10 × 10 × 10 × 10 × 10 = 10^5 = 1,00,000 codes.

Notes

If each choice for one item can go with each choice for another, the number of combinations is the product of the numbers of choices.

Check yourself

Estu has 5 shirts and 2 caps. How many different outfits are possible?

Answer: 10

5 shirts × 2 caps = 10 outfits.

A lock has 3 dials, each with the digits 0 to 9. How many different codes (000 to 999) are possible?

Answer: 1000

10 × 10 × 10 = 1000 codes, from 000 to 999.

A lock has 4 dials, each with the digits 0 to 9. How many codes are possible?

A menu has 3 starters, 4 main dishes and 2 sweets. A meal has one of each. How many different meals can you make?

Answer: 24

3 × 4 × 2 = 24 different meals.

• 4 × 10 = 40. That adds the dials. Every digit of the first dial can go with every digit of the others: multiply 10 four times.
• 10 × 10 × 10 × 10 = 10,000 — correct. Yes! 10 choices for each of the 4 dials: 10^4 = 10,000.
• 10 × 10 × 10 = 1000. That is only 3 dials. There is one more factor of 10.