Digit Permutations

Forming numbers out of digits looks like ordinary permutation counting, but it hides an extra rule: a multi-digit number cannot start with 0. This lesson shows how to handle that leading-digit restriction, and how to combine it with other conditions such as requiring the number to be even or odd.

What you will learn

• Count arrangements of digits into numbers
• Handle leading-digit and repeated-digit restrictions
• Combine place-value reasoning with permutation counting

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

The leading-digit restriction

A 3-digit number cannot begin with 0 — "037" is really just the 2-digit number 37. So whenever 0 is among the digits available for a number-forming problem, the leftmost position has one fewer available choice than the multiplication principle would otherwise suggest, because 0 must be excluded from that position specifically.

This restriction only affects the leftmost position. Once the leading digit is chosen from the allowed nonzero digits, every other position is filled normally from whatever digits remain, including 0 if it has not yet been used. It is a common error to exclude 0 from every position instead of just the leading one, which undercounts the true total.

Fill the most restricted position first

When a problem adds an extra condition on top of "no repeated digits" — a nonzero leading digit, or a units digit that must be even, for instance — figure out how many choices satisfy that specific condition before worrying about the other positions. Then fill the remaining positions with the ordinary multiplication principle, using whatever digits are left after the restricted position is filled.

This order does not change the final count, because the multiplication principle only cares about how many choices exist at each position given everything decided so far — it does not require filling positions from left to right. Filling the most restricted position first just makes it easier to correctly count the choices at every other position, since the restriction has already been accounted for.

When one digit plays two restricted roles

Occasionally a single digit is entangled in two different restrictions at once — for example, 0 is excluded from the leading position, but 0 is also one of the digits that would make a number even. In that situation, whether 0 is used as the units digit changes how many digits remain available for the leading position, so the two restrictions cannot be handled completely independently.

The reliable fix is to split into cases based on whether the doubly-restricted digit is used in its special role: one case where it is (so it cannot also be excluded from the leading position, since it is already placed elsewhere), and one case where it is not (so the leading-digit exclusion applies as usual). Add the counts from each case together. This kind of case split gets a full treatment in a later module, but recognizing when it is needed avoids a subtly wrong count here.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Excluding 0 from every position instead of only the leading position.
• Filling positions left to right by habit even when a different position carries the real restriction.
• Missing that a digit restricted from the leading position can still be entangled with a separate condition, such as being required for a parity condition, which calls for a case split.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A multi-digit number's leading position excludes 0, but every other position may still use 0 once available.
• Decide the most restricted position's choices first, then fill the remaining positions with the multiplication principle.
• If a single digit is entangled in two restrictions at once, split into cases based on where that digit ends up.