How many combinations

Estu has 4 dresses and 3 caps. How many different outfits can he make? With each cap he can wear any of the 4 dresses, so 4 + 4 + 4 = 4 × 3 = 12. Or count it the other way: with each dress there are 3 caps, so 3 + 3 + 3 + 3 = 3 × 4 = 12. Same answer, as it must be.

Now a harder one. Estu and Roxie find their great-grandfather's safe of old stamps and coins, locked with a 5-digit password. Nobody knows it, so they try every possibility, and they're unlucky: it opens only on the very last one. How many did they try?

Rather than guess, take a simpler problem you can solve. The book says so outright: 'If you can't solve a problem, try to find a simpler version of the problem that you can solve. This technique can come in handy many times.'

So try a 2-digit lock. The first digit has 10 options, 0 to 9. For each of those, the second digit has 10 options: if the first is 0, we get 00, 01, 02, …, 09. Total: 10 × 10 = 100.

A 3-digit lock: for each of those 100 two-digit passwords there are 10 choices for the third digit, so 100 × 10 = 1000. You could list them: 000, 001, …, 999.

Now the pattern is clear: each extra slot multiplies by 10. A 5-digit lock has 10 × 10 × 10 × 10 × 10 = 10⁵ = 1,00,000 passwords: just the numbers 00000 to 99999.

Estu says he'll buy a lock with 6 slots of letters A to Z instead, which feels safer. Is he right? Each slot now has 26 choices instead of 10, and there are 6 of them: 26⁶. Work it out and compare.

The same counting is behind codes you meet every day: PIN codes (Vidisha in Madhya Pradesh is 464001, Zemabawk in Mizoram is 796017), mobile numbers and vehicle registration numbers. Find out how each of them is given out.