# Question Video: Evaluating Permutations to Find the Value of an Unknown Mathematics

Find the value of π₯ given that 235ππ₯ β 3π₯ 235π(π₯ β 1) = 0.

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### Video Transcript

Find the value of π₯ given that 235ππ₯ minus three π₯ times 235ππ₯ minus one equals zero.

In order to find π₯, we need to understand the notation used in this equation. The notation πππ means the number of permutations of our unique items taken from a collection of π unique items. And we can calculate this as π factorial divided by π minus π factorial. The factorial of a positive integer π is the product of all the integers from one to π inclusive.

It follows directly from this definition that π factorial is equal to π times π minus one factorial. Equivalently, π factorial is π times π minus one times π minus two factorial and so on. Expanding out factorials like this is very helpful in simplifying expressions.

In our equation, the two elements we need to simplify are 235ππ₯ and 235ππ₯ minus one. Note how similar these are. The only difference is that their π differs by one. When we see something like this, itβs usually the case that one term is a simple multiple of the other term. And this means that we can simplify our expression by figuring out what that multiple is. So, weβre looking in general to find a number such that πππ is that number times πππ minus one. Weβve written the relationship this way because πππ is always greater than or equal to πππ minus one.

Anyway, we can see that the number that weβre looking for is πππ divided by πππ minus one. Expanding using our definition of πππ in terms of factorials, we get π factorial divided by π minus π factorial all divided by π factorial divided by π minus π plus one factorial. π factorial in the numerator of the numerator divided by π factorial in the numerator of the denominator is one. And π minus π factorial in the denominator of the numerator goes into the denominator of the overall fraction. And π minus π plus one factorial in the denominator of the denominator goes into the numerator of the overall fraction.

This leaves us with π minus π plus one factorial divided by π minus π factorial. We can now simplify using our expression for π factorial. This expression tells us that π minus π plus one factorial is equal to π minus π plus one times π minus π factorial. Noting the common factor of π minus π factorial in the numerator and denominator, we see that this expression is simply equal to π minus π plus one. So, the number that weβre looking for in our relationship is π minus π plus one. Now, weβll use this general relationship to substitute in for 235ππ₯ in our original equation.

Replacing π with 235 and π with π₯, we have 235 minus π₯ plus one times 235ππ₯ minus one minus three π₯ times 235ππ₯ minus one equals zero. Now, both terms on the left-hand side have a common factor. Since this common factor is not zero, we can divide both sides of the equation by 235ππ₯ minus one to get rid of it. Our equation then simplifies to 235 minus π₯ plus one minus three π₯ equals zero. 235 plus one is 236, and negative π₯ minus three π₯ is negative four π₯. Adding four π₯ to both sides, we find that 236 is equal to four π₯. And finally dividing both sides by four, we arrive at our answer π₯ equals 59.