Increasing order = just choose: If the answer must be in increasing order, you don't arrange anything — there's only one order.
So you just choose which items to use: that's a selection (ⁿCᵣ).
See it: only one increasing order
Take the digits 7, 2 and 5.
In how many ways can they be written in increasing order?
Step by step
- Smallest first — there's only ONE way.
- So once you've chosen the three digits, the number is fixed — nothing to arrange.
Final answer
Just 1 — increasing order is unique, so you only CHOOSE the digits, never arrange them.
IB-style question — increasing digits
How many 4-digit numbers have strictly increasing digits (for example 1357)?
Step by step
- Choose 4 different digits — each set gives exactly one increasing number. 0 can't be used (it could only lead an increasing number).
- Count the selections.
Final answer
126 numbers.
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Any fixed order works the same way: Alphabetical order (or strictly decreasing) also fixes the order for you — so again you just choose the items.
IB-style question — alphabetical password
A password is made of 3 different letters chosen from A–G, written in alphabetical order.
How many passwords are possible?
Step by step
- Alphabetical order is fixed, so just choose 3 of the 7 letters.
- Count the selections.
Final answer
35 passwords.
Alphabetical order is forced, so the 7 × 6 × 5 ordered picks ÷ 3! → just ⁷C₃ = 35.
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