Animated Solution for Mathematics - Permutations and Combinations: The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is

Unlocking the Combinatorics of Numbers

Imagine you are standing before a set of five distinct digits: $\{3,5,6,7,8\}$. You are tasked with a challenge: form numbers greater than $7000$ without ever repeating a digit.

This is not just a math problem; it is a puzzle of constraints and possibilities. Let us break this down together.

The Anatomy of the Problem

To solve this, we must first recognize that the number of digits in our final integer is the primary filter. A number greater than $7000$ can either have $4$ digits or $5$ digits.

Because we only have $5$ digits available, we cannot form a $6$-digit number. Thus, we split our journey into two distinct, mutually exclusive paths: the $4$-digit path and the $5$-digit path.

The 4-Digit Challenge

Let us visualize four empty slots: [Thousands] [Hundreds] [Tens] [Units]. For the number to be strictly greater than $7000$, the digit in the thousands place is our gatekeeper.

If we place a $3$, $5$, or $6$ there, the number will be in the $3,000$s, $5,000$s, or $6,000$s—all failing our condition. Therefore, the thousands place must be occupied by either $7$ or $8$. This gives us exactly $2$ choices for the first slot.

Once we have locked in either $7$ or $8$, we have used one digit. We are left with $4$ digits to fill the remaining $3$ slots.

This is a classic permutation problem. We need to arrange $4$ items into $3$ positions, which is denoted as ${}^4{P}_3$. Calculating this, we get:

Multiplying our $2$ choices for the thousands place by these $24$ permutations, we find:

The 5-Digit Freedom

Now, consider the $5$-digit numbers. Here, the constraint of being greater than $7000$ vanishes.

The smallest $5$-digit number we can form using our set is $35,678$, which is already far greater than $7000$. Every single permutation of these $5$ digits will result in a number much larger than $7000$.

Since we have $5$ digits and $5$ slots, we are simply arranging all of them. The number of ways to do this is $5!$ (five factorial):

We have successfully navigated both cases. We found $48$ valid $4$-digit numbers and $120$ valid $5$-digit numbers.

Since these cases are mutually exclusive, we simply add them together:

There you have it! By carefully defining our constraints and respecting the rules of permutation, we have arrived at the solution. The total number of valid integers is $168$.