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Hi there. This is a section summary covering the binomial theorem. We can recall that a polynomial that has exactly two terms is called a binomial. We first consider these binomial expansions. We're already familiar with the formulas for expanding the binomial powers x plus y squared and x plus y cubed. In the current section, we can study a method for expanding x plus y to the n-th power for any positive integer n. We can see from these first 5 binomial expansions that as we increase in values of n, the exponent on the binomial expansion, that the number of terms increases greatly. So now, let's look at some properties for expanding x plus y to the n-th power. The patterns for expansions of x plus y to the n-th power where n is equal to 1, 2, 3, 4, and 5, et cetera, suggests the following, that the expansion of x plus y to the n-th power has n plus 1 terms. We also see that the sum of the exponents on x and y in each term equals n. The exponent on x starts at n, or x to the n-th power is equal to a to x to the n-th power times y to the 0-th power in the first term and decreases by 1 for each term until it is 0 in the last term, where x to the 0 times y to the n is equal to y to the n-th power. The exponent on y also starts at 0, where x to the n-th power is equal to a to x to the n-th power times y to the 0-th power in the first term and increases by 1 for each term until it is n in the last term. That is where it's x to the 0 power times y to the n-th power equals y to the n-th power. The variables x and y also have symmetrical roles, that is replacing x with y and y with x in the expansion of x plus y to the n-th power yields the same terms just in reversed order. We also can notice that the coefficients of the first and last terms are both 1. And the coefficients of the second and next to last terms are equal. In general, the coefficients of x to the n minus j times y to the j and x to the j times y to the n minus j are equal for j equals 0, 1, 2, and so on, all the way to n. The coefficients in the expansion of x plus y to the n-th power are called the binomial coefficients. Now, let's look at the coefficients in these binomial expansions for a review of Pascal's Triangle. With x plus y to the 0-th power we have the answer 1. With x plus y to the first power we have 1x plus 1y. With x plus y to the second power, we get 1x squared plus 2xy plus 1y squared. And so on. We can evaluate x plus y to the third power, x plus y to the fourth power and x plus y to the fifth power and we get the following expansions. We can see that the coefficients in these expansions are shown in red text. If we just focus on these coefficients and leave out the variables we have what's called Pascal's Triangle. In Pascal's Triangle we just keep the coefficients in those expansions and so we have ones all along the side here and on the side here. If we add 1 and 1 here, we get 2 that we put right in this position. If we add 1 and 2 here we get 3, which we put here. If we add 2 and 1 here we get 3 here. If we add 3 and 3 here we get 6 here. And so on. So we can, from the triangle, determine all the values of the coefficients for higher and higher values of n. We note the symmetry in Pascal's Triangle. If the triangle were folded vertically down the middle, the numbers on each side of the crease would match. To create a new bottom row in the triangle put the number 1 in the first and last places of the new row and add 2 neighboring entries in the previous row. The top row is called the 0-th row, because it corresponds to the binomial expansion of x plus y to the 0-th power. The next row is called the 1st row, because it corresponds to the binomial expansion of x plus y to the 1st power. All of the rows are names so that the nth throw corresponds to the coefficients of x plus y to the nth power. The coefficients in a binomial expansion can be computed by using ratios of certain factorials. We first introduced the following symbol. If r and n are integers with r greater than or equal to 0, but less than or equal to n, then we define the symbol n choose r to be equal to n factorial divided by r factorial times n minus r factorial. Or n choose 0 is equal to 1 and n choose n is also equal to 1. This symbol is read as n choose r. And it can be shown that n choose r is the number of ways of choosing a subset containing exactly r elements from a set with n elements. The numbers n choose r show up as the coefficients in the expansion of a binomial power. For example, x plus y to the fourth power can be written as either x to the fourth power plus 4x cubed times y plus 6x squared y squared plus 4xy cubed plus y to the fourth power, or 4 choose 0 times x to the fourth power plus 4 choose 1 times x cubed times y plus 4 choose 2 times s squared y squared plus 4 choose 3 times xy cubed and plus 4 choose 4 times y to the fourth power. This result is the binomial theorem for n equals 4, which can be proved by mathematical induction. So the formal definition of the binomial theorem is if n is a natural number, then the binomial expansion of x plus y to the n-th power is given by n choose 0 times x to the n plus n choose 1 times x to the n minus 1 times y plus n choose 2 times x to the n minus 2 times y squared. And so on up to n choose r times x to the n minus r times y to the r plus and so on down to n choose n times y to the nth power, which we can simplify in the following form. That is the sum from r equals 0 to n of n choose r times x to the n minus r power times y to the r-th power. The coefficient of x to the n minus r times y to the r is n choose r, which is equal to n factorial divided by r factorial times n minus r factorial. The binomial theorem provides an efficient method for expanding a binomial power. The binomial theorem can be used to expand a binomial power directly without reference to Pascal's Triangle. This technique is particularly useful in expanding large powers of a binomial. For example, the expansion of x plus y to the 20th power would require you to produce 20 rows of the Pascal's Triangle. The binomial coefficients n choose r are useful in finding a particular coefficient or term in a binomial expansion. We have that a term containing the factor x to the r-th power in the expansion of x plus y to the nth power is n choose n minus r times x to the r-th power times y to the n minus r power. This term also contains the factor y to the n minus r-th power. And so this concludes our section summary on the binomial theorem.