Given that the ratio of 𝑛 plus 19 choose 𝑥 plus 19 to 𝑛 plus 19 choose 𝑥 plus 18 is equal to two to one, determine 𝑛.
So the first thing we’re gonna actually do is to think about it as a proportion. So we’ve got 𝑛 plus 19 choose 𝑥 plus 19 over 𝑛 plus 19 choose 𝑥 plus 18 is equal to two over one. So therefore, the first thing we want to do is actually work out what the value of our first ratio is. But how are we gonna do that?
To help us actually find out what values we have, we can actually use this relationship. And this relationship shows us that if we have a ratio 𝑛 choose 𝑟 to 𝑛 choose 𝑟 minus one, then this is equal to 𝑛 minus 𝑟 plus one to 𝑟.
Okay, so now what are our 𝑛 and our 𝑟 values? Well, our 𝑛 and 𝑟 values are gonna be 𝑛 equals 𝑛 plus 19 and 𝑟 equals 𝑥 plus 19. Okay, so now let’s actually rewrite our ratio using this. So therefore, we can say that 𝑛 plus 19 choose 𝑥 plus 19 to 𝑛 plus 19 choose 𝑥 plus 18 is equal to 𝑛 plus 19 minus 𝑥 plus 19 plus one to 𝑥 plus 19. And this will all possible because when we look back, we can see that our 𝑟 value of our first part in the ratio, so 𝑥 plus 19, is actually one greater than our 𝑟 value in the second part, 𝑥 plus 18. So therefore, we have 𝑟 and 𝑟 minus one.
Okay, so let’s tidy this up. Okay, so we now have the ratio of 𝑛 minus 𝑥 plus one to 𝑥 plus 19. Okay, fab! So now we’ve got this. We’re gonna actually substitute it back into the proportion that we found earlier. So therefore, we can say that 𝑛 minus 𝑥 plus one over 𝑥 plus 19 is equal to two over one.
So now we can actually solve to find our value of 𝑛. So if we cross-multiply, we get 𝑛 minus 𝑥 plus one is equal to two multiplied by 𝑥 plus 19, which is gonna give us 𝑛 minus 𝑥 plus one is equal to two 𝑥 plus 38.
And then finally, if we actually add 𝑥 and subtract one from each side, we’re gonna get that 𝑛 is equal to three 𝑥 plus 37. So therefore, we’ve solved the problem and determined 𝑛.