How Many Number Plates Can a Province Make?

Plates for a tiny province

Six characters make too many plates to list. So start with a make-believe province so small that its plates use only the letters A, B and C and the digits 1 and 2. Build a plate one character at a time, and the widget lists every plate it allows.

Try these, and guess each count before you look:

• one letter: A, B, C, so 3 plates;
• a letter, then a digit: A1, A2, B1, B2, C1, C2, so 6;
• a letter, a letter, a digit: 18;
• three letters: 27.

Look at how the list is laid out. With a letter and then a digit, each first letter starts a row, and every row has the same plates after it: 1 and 2. So there are 3 rows of 2, which is 3 × 2 = 6. Put a letter in front of that, and each of the 3 new first letters starts a row of 6: 3 × 6 = 18.

Back to real plates

Now use all 26 letters and all 10 digits, one character at a time. Each new character multiplies the count by its number of choices.

The plate	Count	Plates
one digit	10	10
two digits	10 × 10	100
four digits	10 × 10 × 10 × 10	10,000
one letter	26	26
two letters	26 × 26	676
two letters, then four digits	676 × 10,000	67,60,000

Four digits make 0000 to 9999, which is exactly the 10,000 numbers you already know. Two letters make 676 pairs, from AA to ZZ. And each of those 676 pairs can go with every one of the 10,000 numbers, so there are 676 rows of 10,000 plates: 67,60,000 (6,760,000 in millions). That is the answer to the opening puzzle, without listing a single plate.

That is the most the format allows. We couldn’t find whether transport offices skip any letters or numbers (0000, say, or the letters I and O, which look like 1 and 0); if they do, the real count is a little smaller. A question at the end works out how much.

Rows of the same length

Why multiply? Draw the choices as a tree. From the start, one branch for each first letter: A, B, C. From each of those, one branch for each digit: 1, 2. Every path from the start to an end is one plate, and there are 3 × 2 = 6 ends.

The tree works because every branch splits the same way. Whichever letter you chose first, you still have the same 2 digits to choose from next. So the 6 plates fall into 3 equal rows of 2, like the grid beside the tree, and counting equal rows is exactly what multiplying does: 3 rows of 2 is 3 × 2, as 3 rows of 2 stones make a rectangle of 6.

Add a third character, and every one of the 6 ends splits again into the same number of branches. Each new character multiplies again.

This is the multiplication principle (also called the product rule for counting or the counting principle):

If a first choice can be made in some number of ways, and, whatever it was, a second choice can always be made in the same number of ways, then the two together can be made in the first number times the second.

It carries on for any number of choices: multiply the numbers of ways for every step. It is the reason you can count 67,60,000 plates without writing them down.

Counting like a meter

The office doesn’t pick plates at random. It gives them out in order, the way a meter turns: AA 0000, AA 0001, AA 0002… up to AA 9999, and then the digits go back to 0000 and the second letter moves on: AB 0000. After AZ 9999 comes BA 0000, and so on to ZZ 9999.

That is just counting, with one difference. In an ordinary number, every place has 10 values, 0 to 9, so each place is worth 10 of the place to its right. On a plate, the digits work the same way, but the letters are places with 26 values:

• one step of the last digit is worth 1 plate; of the digits before it, 10, 100 and 1,000;
• one step of the second letter is worth a whole run of digits: 10,000 plates;
• one step of the first letter is worth 26 runs of 10,000: 2,60,000 plates.

So the plate number tells you the plate, and the plate tells you its number, exactly as with an ordinary number.

Will Bagmati run out?

In the year 2024/25, Nepal registered 1,96,211 new motorcycles and scooters, in all seven provinces together. Even if every one of them had been registered in Bagmati, with plates of class A, the 67,60,000 plates would last

$$67{,}60{,}000 \div 1{,}96{,}211 \approx 34 \text{ years}.$$

And that is one class in one province: cars, buses and the other classes have class letters and plates of their own, and so does every province. By early 2026, Nepal had 62,49,732 registered vehicles of every kind, counting all the years since registration began; that is fewer than one class of one province could number. So the new plates have room to spare.

The old plates

Most plates on Nepal’s roads are still the older kind, painted in Devanagari and read like बा १२ प ३४५६: the zone (बा for Bagmati Zone), a lot number of one or two digits, a letter for the kind of vehicle (प for a private motorcycle), and a number of up to four digits. When a lot’s four-digit numbers are used up, the next lot begins.

How many motorcycles could one zone number? If each lot runs from 1 to 9999 and the lots from 1 to 99:

$$99 \times 9{,}999 = 9{,}89{,}901,$$

just under 10 lakh. And in October 2018, Bagmati Zone got there. Its two-wheeler registrations reached the one-million mark (10 lakh), and new motorcycles and scooters couldn’t be registered: the rules allowed only two digits in the lot number, and the transport department had to wait for the cabinet to allow a third. Here the count wasn’t a puzzle: running out stopped the registration of every new motorcycle and scooter in Bagmati Zone, the Kathmandu Valley included, in the middle of the festival season.

The third digit came. By August 2025 the office at Gurjudhara had registered 15.8 lakh two-wheelers, and it had scanned the files of lots 101 to 114 into its computers. With three digits, the lots go up to 999, and

$$999 \times 9{,}999 = 99{,}89{,}001,$$

almost one crore: one more digit, about ten times as many plates.

The embossed plates are meant to take over, but as of 2026 they are still on their way. The government planned to fit them on every vehicle from September 2025; that same month the plant that makes them was burned in the protests, and in early 2026 the project was still on hold.

“Or” adds, “and then” multiplies

Suppose a province gave out two styles of plate: some with one letter and four digits, others with two letters and three digits. How many plates is that in all?

Each style is counted by multiplying: 26 × 10,000 = 2,60,000 of the first, and 676 × 1,000 = 6,76,000 of the second. But a plate is one style or the other, never both at once, so the two counts are added: 2,60,000 + 6,76,000 = 9,36,000.

That is the companion of the multiplication principle, sometimes called the addition principle: when the things to count fall into separate groups with nothing in two groups, add the groups. And then (a choice, then another) multiplies; or (one group or another) adds.

No digit twice

How many four-digit numbers, 0000 to 9999, use four different digits, like 4071 but not 4041?

The first digit has 10 choices. Whatever it was, the second can be anything but that one: 9 choices. The third has 8 left, the fourth 7. The choices are different digits each time, but their number is always the same, and that is all the multiplication principle needs:

$$10 \times 9 \times 8 \times 7 = 5{,}040.$$

Choosing things one after another, where the order matters and nothing is used twice, is called a permutation. A four-digit PIN with no repeated digit is one of 5,040 permutations; a PIN that may repeat is one of 10,000.

At least one 7

Some people want a plate with a 7 in it, for luck. How many four-digit numbers have at least one 7?

It is tempting to say “the 7 can be in any of 4 places, and the other three digits are anything: 4 × 1,000 = 4,000”. But 7277 would then be counted three times, once for each of its 7s. The way round it is to count the opposite. Numbers with no 7 have 9 choices in each place: 9 × 9 × 9 × 9 = 6,561. Every other number has at least one 7:

$$10{,}000 - 6{,}561 = 3{,}439.$$

Counting what you don’t want and taking it away is often easier than counting what you do.

Where the rule breaks

How many four-digit numbers have their digits going up, each bigger than the one before, like 1357 or 0289?

Try the multiplication principle: 10 choices for the first digit, then…? If the first digit is 0, the second can be any of 9; if it is 5, only 4 (6, 7, 8, 9); if it is 8, only 1, and then there is no room for a third. The number of choices depends on what was chosen before, so there is no single number to multiply by. The principle doesn’t apply.

A smaller case shows what to do instead. Two digits, the second bigger than the first: after 0 there are 9 choices, after 1 there are 8, … after 8 there is 1, after 9 none. The rows have different lengths, so add them: 9 + 8 + 7 + … + 1 = 45.

For four digits there is a neater way. Any four different digits, say 1, 3, 5 and 7, can be put in order in

$$4 \times 3 \times 2 \times 1 = 24$$

ways (4 choices for the first place, 3 for the next…), and exactly one of the 24 goes up: 1357. So the 5,040 numbers with four different digits fall into groups of 24, one group for each set of four digits, and each group has one number going up:

$$5{,}040 \div 24 = 210.$$

The number 4 × 3 × 2 × 1 is written 4! and read “4 factorial”: the number of ways to put 4 different things in a row. Choosing a group of things where the order doesn’t matter (just the four digits, not which comes first) is called a combination. There are 210 combinations of 4 digits out of 10, and 5,040 permutations.

• How Many Handshakes at a Party of 30? Counting pairs, where the order doesn’t matter, three ways that all agree.
• How Many Ways Can You Walk to School? Routes along a grid of lanes, counted with Pascal’s triangle.
• How Much Rice Is on the Chessboard? Multiplying by the same number again and again, and how fast it grows: the same growth as each new character on a plate.
• Will Two People in Your Class Share a Birthday? (Grades 9–12) The multiplication principle turned into chances, and a surprising answer.
• How Many Ways Round Bhaktapur’s Temples? (Computing, Going Further 6–8) Five temples have only a few round trips; twenty have more than any computer could ever try.
• What Is the Biggest Number a Computer Can Count To? (Computing, Going Further 6–8) A counter with a fixed number of places, like a plate, and what happens when it reaches the end.
• Grade 10 Math: Multiplication principle of probability The same principle, used for chances.