How does binomial theorem work

This pattern continues indefinitely. A simple construction of the triangle proceeds in the following manner.

Then, to construct the elements of following rows, add the two above numbers to find the new value. If either of the above numbers is not present, substitute a zero in its place. Applying these numbers to the binomial expansion, we have:.

The theorem is given by the formula:. The coefficients that appear in the binomial expansion are called binomial coefficients. The formula consists of factorials:. Sections: The formulas, Worked examples. The Binomial Theorem is a quick way okay, it's a less slow way of expanding or multiplying out a binomial expression that has been raised to some generally inconveniently large power. The formal expression of the Binomial Theorem is as follows:.

Yeah, I know; that formula never helped me much, either. And it doesn't help that different texts use different notations to mean the same thing. The parenthetical bit above has these equivalents:. Recall that the factorial notation " n! Then the notation " 10 C 7 " often pronounced as "ten, choose seven" means:. Many calculators can evaluate this " n choose m " notation for you. Just look for a key that looks like " nCm " or " nCr ", or for a similar item on the "Prob" or "Math" menu, or check your owner's manual under "probability" or "combinations".

The evaluation will probably look something like this:. Previous Sequences and Series. Removing book from your Reading List will also remove any bookmarked pages associated with this title. Are you sure you want to remove bookConfirmation and any corresponding bookmarks? My Preferences My Reading List. You can read more at Combinations and Permutations. The handy Sigma Notation allows us to sum up as many terms as we want:. Sigma Notation. And let's not forget "8 choose 5" We can use the Binomial Theorem to calculate e Euler's number.

Try calculating more terms for a better approximation! Try the Sigma Calculator.