PREVNEXTSUBMITMenuResourcesGlossaryNotesClear and return to MenuSearch ResultsFilterSlide TextNotesSearch in:Check to includeSearch...TermsDefinitiondefinitionresourcessearchpauseplayreplaysubmitnextpreviousvolumeproblem solving with pascal's triangle example 1: coins can be arranged in the shape of an equilateral triangle as shown: continue the pattern to determine the number of coins in triangles with four, five and six rows. how many coins are in a triangle with 12 rows? example 1: solution: (a). the number of coins in the triangle follow the pattern 1 + 2 + 3 + …. as shown below: (b). a triangle with 12 rows contains 78 coins. example 2: george polya was a mathematician and mathematics teacher. he dedicated to the problem-solving approach in the teaching of mathematics. in his words, “there you may experience the tension and enjoy the triumph of discovery.” george polya in the arrangement of letters given starting from the top we proceed to the row below by moving diagonally to the immediate right or left. how many different paths will spell the name george polya? click on the button to learn more about example 2: solution: we determine the number of paths leading to each letter in the arrangement as shown using pascal's triangle. completing this, we see there are 120 ways to reach the bottom; therefore, there are 120 different paths that will spell george polya. example 3: determine how many different paths will spell pascal, if you start at the top and proceed to the next row by moving diagonally left or right. solution: starting from the top, record the number of possible paths moving diagonally to the left or right as you proceed to each different letter. for instance, there is one path from an a to the left s, two paths from an a to the middle s, and one path from an a to the right s. continuing with this pattern, we determined a total of 20 different paths that spell pascal. example 4: on the checkerboard shown, the checker can travel only diagonally upward. it cannot move through a square containing an x. determine the number of paths from the checker's current position to the top of the board. click here to learn more about the game checkers click here to play the online game of checkers example 4: solution: we can use pascal's method to find the number of paths to each successive position. there is one path possible into each of the squares diagonally adjacent to the checker's starting position. from the second row there are four adjacent paths to the third row; one path to the third square from the left, two to the fifth square, and one to the seventh square. continue this process for the remaining four rows. the square containing an x gets a zero or no number since there are no paths through this blocked square. from left to right, there are 5, 9, 8, and 8 paths to the white square at the top of the board, making a total of: 5 + 9 + 8 + 8 = 30 paths. a useful and surprising application of pascal's triangle is in the expansion of binomials. in your previous mathematical experience, we can expand and collect the like terms of (a + b)2 as: (a + b)2 = (a + b)(a + b) (a + b)2 = a2 + 2ab + b2 for higher powers of (a + b), the expansion would proceed as: (a + b)3 = (a + b)(a + b)2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = (a + b)2(a + b)2 (a + b)4 = a4 + 4a3b + 6a2b2 +4ab3 + b4 unfortunately, the use of this method to expand (a + b)12 requires the previous expansion of (a + b)11, which in turn requires knowing the expansion of (a + b)10, and so on. to discover how pascal's triangle can be useful in expanding higher powers, let us summarize the expansions we have already determined in this section. (a + b)0 and (a + b)1 are also included. value of n (a + b)n 0 (a + b)0 = 1 1 (a + b)1 = a + b 2 (a + b)2 = a2 + 2ab + b2 3 (a + b)3 = a3 + 3a2b + 3ab2 + b3 4 (a + b)4 = a4 + 4a3b + 6a2b2 +4ab3 + b4 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 first, notice that all expansions proceed in decreasing powers of a and increasing powers of b, their exponents always adding to the exponent on the binomial. second, the coefficients are precisely the numbers in pascal's triangle. example 5: use pascal's triangle to determine the expansion of (a + b)6 . solution: extending pascal's triangle for n = 6, we obtain the following: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 example 5: proceed with decreasing powers of a and increasing powers of b, we get: (a + b)6 = 1a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + 1b6 1 6 15 20 15 6 1