### Video Transcript

If 49π π plus three equals 34
times 49π π plus two, find the value of π minus six factorial.

One way to solve this will be to
first convert each of these permutations into factorial form using the definition
πππ equals π factorial over π minus π factorial. We rewrite 49π π plus three as 49
factorial over 49 minus π plus three factorial. Be careful to notice that in this
case, our π, the number of elements weβre choosing, is π plus three and that the
subtraction applies to both the π and the three. The other side of the equation
becomes 34 times 49 factorial over 49 minus π plus two factorial. We have 49 factorial in the
numerator on both sides of the equation, which means we can multiply both sides of
the equation by one over 49 factorial.

On the left, weβll be left with the
numerator of one. And in the denominator, we can
subtract π and subtract three from 49, which will give us one over 46 minus π
factorial. The numerator on the right side
becomes 34, and the denominator is 47 minus π factorial. At this point, it might not seem
clear what we should do. But since weβre trying to solve for
π, itβs a good idea to get both π-values on the same side of the equation. So weβll multiply both sides of the
equation by 47 minus π factorial. This gives us 47 minus π factorial
over 46 minus π factorial equals 34. Our denominator is a factorial that
is one less than our numerator.

We know that π factorial is equal
to π times π minus one factorial. We can apply this here, but we need
to be careful. In our case, the π is 47 minus
π. And that means 47 minus π
factorial is equal to 47 minus π times 47 minus π minus one factorial. We can simplify 47 minus π
factorial to be equal to 47 minus π times 46 minus π factorial. If we substitute this in for our
numerator, the 46 minus π factorial cancels out in the numerator and the
denominator. We then have 47 minus π equals
34. If we subtract 47 from both sides,
we find that negative π equals negative 13, which makes π equal to positive
13. Our final step will be to calculate
π minus six factorial, which we now know is 13 minus six factorial. Solving this by hand or with a
calculator will yield that seven factorial equals 5,040.