Why 5! equals 120: a friendly look at factorials in HSPT math

Outline (brief skeleton)

• Hook and context: factorials show up in counting and order; this one is simple but powerful.

• Quick primer: what is a factorial and what does n! mean?

• Step-by-step: how to compute 5! with clear multiplication steps.

• Why it matters: factorials explain how many ways to arrange objects (permutations).

• Real-world anchor: a friendly, everyday analogy to make the idea click.

• Practical tips for HSPT-style questions: how to spot factorials, how to do mental math fast.

• The moment of truth: applying the idea to the given question; identifying the correct answer and why.

• Final takeaways: a loose guide for recognizing factorials in math problems.

What’s the big idea behind factorials?

Let me explain with a simple question in mind: you have five distinct objects, and you want to line them up in a row. How many different ways can you arrange them? This is exactly the kind of problem factorials are built for. The notation n! (read as “n factorial”) means you multiply all positive integers from 1 up to n. So, 5! means the product 5 × 4 × 3 × 2 × 1. That’s the core trick you’ll see again and again in math problems, including ones that pop up on the HSPT.

What exactly is a factorial?

A factorial is just a tidy way of saying “count all the ways to order n distinct items.” If you have n items, there are n choices for the first position, then (n−1) choices for the second, and so on, down to 1 choice for the last position. Multiply all those choices together, and you’ve got n!. It’s a compact shorthand for a very common counting pattern. Easy to remember, but with a lot of power.

Step-by-step: calculating 5!

So, let’s work out 5! in a clean, straightforward way. You multiply all the numbers from 1 to 5:

5! = 5 × 4 × 3 × 2 × 1

Now, a quick, practical breakdown:

• Start with 5 × 4 = 20.

• Then 20 × 3 = 60.

• Next, 60 × 2 = 120.

• Finally, 120 × 1 = 120.

That last step—multiplying by 1—really doesn’t change the number, but it’s part of the full product. When you finish, you’ve got 120. In other words, 5! equals 120.

Why this number matters beyond the arithmetic

The idea behind 5! isn’t just “multiply, multiply, multiply.” It’s about counting arrangements. If you have five different objects (say, five different colored balls or five distinct books), there are 120 different orders you could arrange them in a row. That repetition-free, clear sense of “how many orders?” is what factorials deliver. It helps you see why 5! is so much bigger than, say, 5 × 4, or 5 × 4 × 3, or any smaller chunk—you’re counting every possible lineup, not just a single product.

A friendly analogy to anchor the concept

Imagine you’re seating five friends at a long table. The first seat could be any of the five friends. Once you’ve picked who sits first, there are four people left for the second seat, then three for the third, and so on. You’re multiplying choices at each step, and that product is 5!. It’s the kind of mental model that helps you glance at a problem and see, “Oh—this is about orderings, not just numbers.” The same mentality shows up in many HSPT-style questions, sometimes in disguise.

Practical tips for spotting factorials on questions

• Look for the exclamation mark after a number. That little symbol is the signal flag saying “multiply down to 1.”

• If the problem mentions arranging items or counting different orders, factorials are often lurking behind the scenes.

• For mental math,’multiplying by 5, then by 4, then by 3, etc., is a predictable pattern. Practice a few quick checks in your head to avoid mistakes.

• Remember the shortcut: n! grows fast. Don’t panic if you see a large product—break it down step by step, like we just did.

A real-world touchstone: permutations in everyday life

Here’s a quick, relatable scenario: you’ve got five different puzzle pieces, and you want to know how many distinct ways you can lay them out on the table. That count is 5!. It isn’t just a brain teaser; it’s a window into why certain problems feel intuitive while others require a careful, methodical approach. The same logic appears in test questions about order, sequences, and arrangements.

How to approach factorial questions on this kind of content

• Start with the definition in your head: n! means multiply all integers from 1 to n.

• Map the problem to an arrangement idea: are you counting how many orders? If yes, factorials are likely involved.

• Do a quick left-to-right multiplication, pausing to verify each step. Small checks help prevent misplaced digits.

• Use tools sparingly and smartly: a quick calculator or a trusted math app (think Wolfram Alpha, Desmos, or a calculator app) can confirm if your mental arithmetic trips you up.

• If the problem gives you several options, backtrack by dividing the option by smaller factors to see if you land on a neat chain back to 1.

The moment of truth: applying the concept to the given question

Let’s bring the idea home with the specific problem:

Question setup:

What is the value of 5! (5 factorial)?

A. 60

B. 120

C. 100

D. 150

Here’s the thing: you’re being asked to evaluate the factorial directly. As we walked through, 5! is the product 5 × 4 × 3 × 2 × 1. Recalling the steps:

• 5 × 4 = 20

• 20 × 3 = 60

• 60 × 2 = 120

• 120 × 1 = 120

So the value collapses to 120. The answer is B.

That conclusion isn’t just a number. It’s a demonstration of a pattern you can reuse: whenever you see n!, you’re counting every possible order of n distinct items. And with 5 items, there are 120 different orders. If a question gives you similar options, you can approach them with the same steady workflow—step by step, no rush, and always check your work.

A few closing reflections you can carry forward

• Factorials aren’t exotic; they’re a natural extension of counting. The more you see them, the more you start to recognize when a problem is about order and not just a single arithmetic result.

• Keep a little mental checklist: identify the factorial, recall the product pattern, and verify by recomputing in small steps if you’re unsure.

• Don’t rush the multiplication. Slow, steady steps are your friends here. A tiny error in a single step can throw off the whole result.

• When you’re curious, use a real-world analogy to sanity-check your answer. If you can seat items or people in many orders, you’re likely dealing with a factorial.

Where to go from here, in plain terms

If you want to keep this momentum, try a few simple exercises with the same idea. Take five items—be they pencils, bracelets, or stickers—and pretend you’re lining them up. Count how many different orderings you can create. Then try 4! or 6! to see the pattern. You’ll notice the numbers aren’t random—they’re the fruit of a universal counting rule.

And if you’re the kind of learner who likes to hear the math thoughts out loud, a quick chat with a calculator or a math app can help you confirm your steps without feeling like you’re wrestling with a wall. It’s all about building confidence, little by little, until the ideas feel as natural as tying your shoes.

Bottom line

5! equals 120, and that number isn’t just a value on a line of text. It’s a doorway into a way of thinking about order, choice, and counting. When you see a factorial on a problem, you’re looking at how many different sequences you can create with n distinct items. With practice, these ideas start to click faster, and the correct answer almost arrives before you finish the first sweep.

If you’re curious about more examples, or you want to see how factorials show up in slightly bigger numbers, I’m happy to walk through a few more scenarios. The technique stays the same, and the confidence grows with each one. After all, math is less about memorizing isolated facts and more about recognizing the patterns that quietly run the ship beneath the numbers.