If ¡n! = (n!)2, then ¡17! – ¡16! =
Correct Answer: ¡16!)(122)(2)
My Answer: ¡1!
I've read the Manhattan prep Atlas explanation about 5x now, and I still don't get it. Please help me understand this, and explain this to me like I'm 5-years-old.
Thanks for your time!
Manhattan Prep Atlas Explanation:
The problem defines a made-up symbol; this type of problem is essentially asking you to follow directions. The weird symbol means a sort of “double factorial”: take the factorial and then take it again! For example, ¡5! would be (5)(4)(3)(2)(1)(5)(4)(3)(2)(1).
¡17! and ¡16! would work the same way as ¡5!, but nobody would want to multiply those out without a calculator. Is there a shortcut? Glance at the form of the answers—most still use the weird upside-down-exclamation symbol. There must be a way, then, to rewrite ¡17! and ¡16! in another form.
There is! The two numbers are close to identical, but ¡17! contains two additional factors: 17 and 17. Factor out a ¡16! from the two terms:
¡17! – ¡16! =
¡16! (17×17 – 1)
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Re: CAT 6: Spanish Factorials
Apologies for the delay in replying to your question.
This is a pretty devilish problem, so it's pretty usual to be confused by it. The first challenge is to understand the idea of 'made up functions' that you find on GMAT problems. A simpler example is PS 196 from OG 2020. There are conventional mathematical symbols that you learn about, such as the + symbol. When you see this symbol between two numbers, you know to add the numbers together. Mathematicians sometimes make up new symbols to suit their purposes, and GMAT tests our logic by doing so too. Here's a simple example: let's say that x@y means "multiply x and y, then add y". So that 4@5 would equal 25. What would 3@2 be? Answer: 8 We call these problems formula problems, and there's a good chapter about them in All the Quant.
Applying that to this problem, we have the 'made up function' ¡5! = (1)(2)(3)(4)(5)(5)(4)(3)(2)(1), so that ¡17! = 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17 and ¡16! =1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16 Phew!
So, ¡17! – ¡16! = 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17 – 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16
We could notice that there are lots of common factors, so we can factorize it:
¡17! – ¡16! = (17x17 – 1)(1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16)
And then notice that 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16 = ¡16!
So that ¡17! – ¡16! = (17x17 – 1)(¡16!)
Finally, 17x17 – 1 = 389 – 1 = 288 = 12x12
If that looks overwhelming, then be aware that I certainly wouldn't write all that out in an exam. This technique of factorizing is one that is often used in problems involving the factorial function (e.g. 4! + 3! = (4 + 1) 3!).
This is a pretty devilish problem, so it's pretty usual to be confused by it. The first challenge is to understand the idea of 'made up functions' that you find on GMAT problems. A simpler example is PS 196 from OG 2020. There are conventional mathematical symbols that you learn about, such as the + symbol. When you see this symbol between two numbers, you know to add the numbers together. Mathematicians sometimes make up new symbols to suit their purposes, and GMAT tests our logic by doing so too. Here's a simple example: let's say that x@y means "multiply x and y, then add y". So that 4@5 would equal 25. What would 3@2 be? Answer: 8 We call these problems formula problems, and there's a good chapter about them in All the Quant.
Applying that to this problem, we have the 'made up function' ¡5! = (1)(2)(3)(4)(5)(5)(4)(3)(2)(1), so that ¡17! = 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17 and ¡16! =1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16 Phew!
So, ¡17! – ¡16! = 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17 – 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16
We could notice that there are lots of common factors, so we can factorize it:
¡17! – ¡16! = (17x17 – 1)(1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16)
And then notice that 1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16 = ¡16!
So that ¡17! – ¡16! = (17x17 – 1)(¡16!)
Finally, 17x17 – 1 = 389 – 1 = 288 = 12x12
If that looks overwhelming, then be aware that I certainly wouldn't write all that out in an exam. This technique of factorizing is one that is often used in problems involving the factorial function (e.g. 4! + 3! = (4 + 1) 3!).
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