Question
What is the Pascal's Triangle? How does it work? Thanks for taking time to answer this question.
Answer
Pascal's triangle is a particular table of numbers in the shape of a triangle. The first row of the triangle has 1 number, the next has 2 numbers, and so on. The first few rows of Pascal's triangle are<BR> 1<BR> 1 1<BR> 1 2 1<BR> 1 3 3 1<BR> 1 4 6 4 1<BR>Each entry in Pascal's triangle is the sum of the two entries just above it, except for the entries on the edge, which are all 1.<BR><BR>You might be able to see that each entry in the triangle is the number of downward paths to that entry from the top 1, where a path jogs downwards from row to row, going from an entry to one of the two nearest entries in the row below. To see this, notice that every path to an entry must go through one of the two entries just above it, so each path to one of those two entries can be extended (in exactly one way) to a path to the entry in question.<BR><BR>You may also be able to see that if we number the rows starting at 0, and the entries in each row starting a 0, that the nth entry in row r is the number of ways to pick n items from a list of r items. For example, the first 6 in the triangle is the number of ways to pick 2 items from a list of 4. To see this, think of each of the rows 0,1,2,...r-1as representing one of the items, and each path from the top of the triangle to the nth entry in row r as specifying the choices, where a jog to the left from a row means the item corresponding to that row is not picked, an a jog to the right means the item is picked. To get to the nth item in row r, we have to jog right exactly n times, meaning exactly n items are picked (namely the n items corresponding to the rows where the path jogged right).<BR><BR>For this reason, the nth item in the rth row is called "the number of combinations of r things taken n at a time" or "r choose n", sometimes written rCn. So 4C2 = 6. These numbers are also called "binomial coefficients", because (1 + x)<SUP>r </SUP>= rC0 + rC1 x + rC2 x<SUP>2</SUP> + rC3 x<SUP>3</SUP> + ... + rCr x<SUP>r</SUP>. To see this, notice that (1+x)<SUP>r</SUP> is (1+x) multiplied by itself r times. For each factor of (1+x), we can pick either the 1 or the x to multiply. Just think of a path in Pascal's triangle from the top to row r as representing the choices of 1 (jog left) or x (jog right). By the way, this immediately leads to the observation that the sum of the entries in row r is 2<SUP>r</SUP>.<BR><BR>There are many other interesting properties of the entries in Pascal's triangle. See some of the URLs below. There are also other triangles similar to Pascal's. For example, if instead of just adding the two entries we first multiplied the right one by its position in its row, we'd get a table of "Stirling numbers of the second kind", which give the number of ways of grouping r items into n nonempty sets. This triangle looks as follows:<BR>1<BR>0 1<BR>0 1 1<BR>0 1 3 1<BR>0 1 7 6 1<BR>0 1 15 25 10 1<BR><BR>The field of mathematics in which Pascal's triangle finds itself is called "combinatorics".<BR><BR><BR><BR><BR><BR> http://ptri1.tripod.com/http://mathforum.org/dr.math/faq/faq.pascal.triangle.htmlhttp://mathworld.wolfram.com/PascalsTriangle.html Pascal<BR>binomial<BR>coefficient<BR>combination<BR>combinatorics<BR> http://vrd.askvrd.org/services/answerschema.xml
