How many 5-digit numbers have their digits arranged in strictly increasing order?

The exact number of 5-digit numbers with their digits arranged in strictly increasing order is 126.

Understanding Strictly Increasing 5-Digit Numbers

A 5-digit number must have its first digit non-zero. For digits to be arranged in strictly increasing order, each subsequent digit must be larger than the one preceding it. For example, 12345 or 24679.

Key Characteristics:

• No Zero Allowed: If any digit in a strictly increasing sequence were zero, it would have to be the smallest digit. However, a 5-digit number cannot start with zero (e.g., 01234 is a 4-digit number). If zero appeared elsewhere, say d2=0, then d1 would have to be less than zero, which is not possible for a digit. Therefore, all digits must be chosen from the set of non-zero digits: {1, 2, 3, 4, 5, 6, 7, 8, 9}.
• Unique Arrangement: When you select a set of distinct digits, there is only one way to arrange them in strictly increasing order. For instance, if you choose the digits {2, 5, 7, 8, 9}, the only 5-digit number you can form with these digits in strictly increasing order is 25789. This means that finding the number of such numbers is equivalent to finding the number of ways to choose the digits themselves.

The Combinatorial Approach

Since we need to select 5 distinct digits from the 9 available non-zero digits ({1, 2, 3, 4, 5, 6, 7, 8, 9}), and the order of selection does not matter (as they will automatically be arranged in increasing order), this is a classic combination problem.

The number of ways to choose k items from a set of n items (where order does not matter) is given by the combination formula:

C(n, k) = n! / (k! * (n-k)!)

In this case:

• n = 9 (the total number of non-zero digits available)
• k = 5 (the number of digits required for the 5-digit number)

Calculation:

C(9, 5) = 9! / (5! (9-5)!)
= 9! / (5! 4!)
= (9 × 8 × 7 × 6 × 5!) / (5! × 4 × 3 × 2 × 1)
= (9 × 8 × 7 × 6) / (4 × 3 × 2 × 1)
= (3024) / (24)
= 126

This calculation confirms that there are exactly 126 such numbers.

Examples of Such Numbers:

Here are a few examples of 5-digit numbers with strictly increasing digits:

• 12345
• 12346
• 12347
• ...
• 12349
• 12356
• ...
• 56789

Summary Table

Mathematics, Combinatorics