Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. The binomial theorem definition of a binomial coefficient is n / r or N above R&N above r = n factorial divided by R factorial times the quantity n -, r factorial. Remember that zero factorial really just equals 1A. Formula for expanding binomials. The binomial theorem For any positive integer NA plus B to the north power equals N above R or N above 0A to the north plus N above 1A to the north -1 B plus N above 2A to the north minus two b ^2 plus N above 3A to the north -3 B to the third plus... dot dot plus N to the north above north of B to the north. Now something to look at is each of the coefficients of each term should add up to the north. If we add the coefficients, each of those terms should be the same as whatever the power was originally finding a particular term in a binomial expansion, the r + 1 term of the expansion of A+B to the north is N above RA to the north minus r * b to the R. Let's look at some examples. Oh, actually, let's not look at examples yet. Pascal's triangle is an array of numbers showing coefficients of the term. So Pascal's triangle starts with ones, and when we come out we put ones on the outside and then if we have terms we add the two that are together. So one and one is 2/1 and two is 3, two and one is 3-1 and three is 4/3 and three and six. Three and one is 4-1 and four are 5 four and 6:10 six and 4:10 four and 1/5. This is another way of finding the coefficients. So if you looked here, we have a coefficient of 1510105 and one, which would be the coefficients on the 5th row out. So it's the same thing as doing the N above our concept. If I have seven above 2, that's seven factorial divided by two factorial times the quantity 7 - 2 factorial, 7 factorial over two factorial times 5 factorial, 7 * 6 factorial over 2 * 1 * 5 factorial 7 * 6 / 2, which is 20. One 5 / 0 would be 5 factorial over 0 factorial times the quantity 5 - 0 factorial. That simplifies up to just being one 7 / 7 is 7 factorial divided by 7 factorial times 7 - 7 factorial. That one also simplifies to be 1/8 over 38 factorial over three factorial times 8 - 3 factorial. So 8 * 7 * 6 * 5 factorial. The three factorial is going to be 3 * 2 * 1 and the five factorial I'm going to leave so that they can cancel on the numerator and denominator 3 * 2 is 6 so that's going to reduce and leave us just 8 * 7 or 56. The next example, if we have X + 5 all to the 4th, using the formula that says 4 / 0 X to the 4th plus four over One X ^3, 5 to the first plus 4 / 2 X squared 5 ^2 + 4 / 3 X 5 ^3 + 4 / 4 five to the 4th. Remember, when we look at the exponents total, they have to be the same thing as the original powers. So three and 1/2 plus two 1 + 3. There's the four. So 4 factorial over 0 factorial. 4 factorial is just 1 * X to the 4th 4 factorial over one factorial times 4 -, 1 factorial. It's really 4 factorial over three factorial, which is just plain old 4. Continuing in that fashion, we end up with X to the 4th plus twenty X ^3 + 1 fifty X ^2 + 5 hundred X + 625. Now, you could have done Pascal's Triangle here if you had wanted. We would have had to have gone out to the 4th location and we would have seen that the coefficients were 1464 and one. There's the one, there's the four, there's the six, there's the four, and there's the one. We still have to multiply that by that five to the various powers. Look at another example. We have the quantity 2 X -3 Y to the 5th. So we're going to have 5 / 0. Oops, 5 / 0 two X to the fifth plus five over 12X to the fourth times -3 Y to the first. The negative does need to go along. Plus 5 / 2 two X ^3 * -3 Y quantity squared plus 5 / 3, etcetera. When we do this one, we end up with 32X to the 5th -240 X to the fourth times y + 720 X cubed y ^2 -1000 eighty X ^2 y ^3 + 810 XY to the 4th -243 Y to the fifth. Once again, you could have done Pascal's triangle. If you do Pascal's triangle, you'll see that the coefficients are 1510105 and one, and that's before we multiply it through by the coefficients that were inside our last example. If it tells us to find the fifth term of two X + y to the 9th power, we're going to use the formula. That said, the R Plus first term is really just n / r times the A to the N minus RB to the R. So if we wanted the fifth term, we're going to say r + 1 equal 5, so R is 4. We know that it's nine for N because that's the power. So 9 / 4 two X to the 9 -, 4 * y to the four. So 9 / 4 is 9 factorial over 4 factorial. 9 -, 4 factorial 2X to the fifth Y to the 4th, 9 factorial over 4 factorial, 5 factorial times 32 X to the fifth Y to the 4th. If I do my 9 factorial down to five factorial, I'm using the highest one to stop at over my 4 factorial times 5 factorial. The five factorials are going to cancel 4 * 2. Is that 8? So that eight goes away? 3 went into six twice so I end up with 9 * 763 * 2 so 126 * 32 X to the fifth Y to the 4th or 4032 X to the fifth Y to the 4th. Thank you and have a wonderful day.