Understanding Combinations and Permutations: Are They Equal?

Is ⁶C₁ equal to ⁶P₁?

• A. Yes, equals 6
• B. No
• C. Yes, equals 1
• D. Only for larger numbers

Answer: (A)

To understand whether ⁶C₁ is equal to ⁶P₁, it's crucial to look at what these notations represent. ⁶C₁ denotes the number of combinations of 6 items taken 1 at a time. The formula for combinations is given by: \[ ⁿCᵣ = \frac{n!}{r!(n-r)!} \] In this case, we have: \[ ⁶C₁ = \frac{6!}{1!(6-1)!} = \frac{6!}{1! \times 5!} = \frac{6 \times 5!}{1 \times 5!} = 6 \] This means there are 6 ways to choose 1 item from a set of 6. ⁶P₁ represents the number of permutations of 6 items taken 1 at a time. The formula for permutations is given by: \[ ⁿPᵣ = \frac{n!}{(n-r)!} \] For this case, we have: \[ ⁶P₁ = \frac{6!}{(6-1)!} = \frac

When it comes to tackling GMAT questions about combinations and permutations, understanding whether (⁶C₁) equals (⁶P₁) is essential. Let's break this down together—no need for a calculator, just your trusty brain (and maybe a snack!) to keep you fueled.

You see, (⁶C₁) is the number of combinations of 6 items taken 1 at a time, and it has a formula:

[

ⁿCᵣ = \frac{n!}{r!(n-r)!}

]

Plugging in our numbers gives:

[

⁶C₁ = \frac{6!}{1!(6-1)!} = \frac{6!}{1! \times 5!} = \frac{6 \times 5!}{1 \times 5!} = 6

]

What this means is there are 6 ways to select 1 item from a set of 6—imagine deciding which snack to grab from a bowl of six yummy treats. That's simplicity at its best, right?

Now, let’s compare this with permutations. (⁶P₁) represents the number of ways we can arrange 1 item out of 6, which isn't fundamentally different from combinations in this case. The formula for permutations is:

[

ⁿPᵣ = \frac{n!}{(n-r)!}

]

Using this for our scenario, we get:

[

⁶P₁ = \frac{6!}{(6-1)!} = \frac{6!}{5!} = 6

]

Again, we find the answer is 6. So, in this particular instance, yes—(⁶C₁) does indeed equal (⁶P₁)!

This might surprise some folks. I mean, you might be wondering why these two seemingly similar concepts yield the same result when we're just picking one item. The magic lies in the simplicity; whether you’re choosing or arranging—if it’s just one, there’s only one choice in the end.

But hey, if you were selecting more than one item, the story changes dramatically. If you were asked to select 2 items from our original 6, you'd see a different picture. The number of combinations becomes more complex, but that’s for another chat!

For now, here's a pro tip: understanding the differences between combinations and permutations—especially their formulas—is crucial. Make sure you're comfortable with both, as it'll help you breeze through various GMAT problems.

Use practice problems that force you to switch gears between combinations and permutations; after all, living in your comfort zone won't prepare you for the unexpected twists that the GMAT throws your way. Familiarize yourself with sample questions or online resources that focus on these mathematical concepts, and remember—practice makes perfect.

So, as you navigate your GMAT journey, keep this intriguing bit of information tucked neatly away in your mind. Feeling confident? You should be! It's little revelations like this that add up to a successful study experience.