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