Find the value of 𝑥 given that 235𝑃𝑥 minus three 𝑥 times 235𝑃𝑥 minus one equals zero.
In order to find 𝑥, we need to understand the notation used in this equation. The notation 𝑛𝑃𝑟 means the number of permutations of our unique items taken from a collection of 𝑛 unique items. And we can calculate this as 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. The factorial of a positive integer 𝑛 is the product of all the integers from one to 𝑛 inclusive.
It follows directly from this definition that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial. Equivalently, 𝑛 factorial is 𝑛 times 𝑛 minus one times 𝑛 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 235𝑃𝑥 and 235𝑃𝑥 minus one. Note how similar these are. The only difference is that their 𝑟 differs by one. When we see something like this, it’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, we’re looking in general to find a number such that 𝑛𝑃𝑟 is that number times 𝑛𝑃𝑟 minus one. We’ve written the relationship this way because 𝑛𝑃𝑟 is always greater than or equal to 𝑛𝑃𝑟 minus one.
Anyway, we can see that the number that we’re looking for is 𝑛𝑃𝑟 divided by 𝑛𝑃𝑟 minus one. Expanding using our definition of 𝑛𝑃𝑟 in terms of factorials, we get 𝑛 factorial divided by 𝑛 minus 𝑟 factorial all divided by 𝑛 factorial divided by 𝑛 minus 𝑟 plus one factorial. 𝑛 factorial in the numerator of the numerator divided by 𝑛 factorial in the numerator of the denominator is one. And 𝑛 minus 𝑟 factorial in the denominator of the numerator goes into the denominator of the overall fraction. And 𝑛 minus 𝑟 plus one factorial in the denominator of the denominator goes into the numerator of the overall fraction.
This leaves us with 𝑛 minus 𝑟 plus one factorial divided by 𝑛 minus 𝑟 factorial. We can now simplify using our expression for 𝑛 factorial. This expression tells us that 𝑛 minus 𝑟 plus one factorial is equal to 𝑛 minus 𝑟 plus one times 𝑛 minus 𝑟 factorial. Noting the common factor of 𝑛 minus 𝑟 factorial in the numerator and denominator, we see that this expression is simply equal to 𝑛 minus 𝑟 plus one. So, the number that we’re looking for in our relationship is 𝑛 minus 𝑟 plus one. Now, we’ll use this general relationship to substitute in for 235𝑃𝑥 in our original equation.
Replacing 𝑛 with 235 and 𝑟 with 𝑥, we have 235 minus 𝑥 plus one times 235𝑃𝑥 minus one minus three 𝑥 times 235𝑃𝑥 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 235𝑃𝑥 minus one to get rid of it. Our equation then simplifies to 235 minus 𝑥 plus one minus three 𝑥 equals zero. 235 plus one is 236, and negative 𝑥 minus three 𝑥 is negative four 𝑥. Adding four 𝑥 to both sides, we find that 236 is equal to four 𝑥. And finally dividing both sides by four, we arrive at our answer 𝑥 equals 59.