Pascal's Triangle Concepts and Applications

Pascal's Triangle, known by various names across cultures, is a mathematical structure full of patterns and secrets. It is generated by adding numbers in pairs, and each row corresponds to the coefficients of a binomial expansion. The triangle reveals patterns like powers of two and geometric shapes such as triangular and tetrahedral numbers. It also has applications in probability and combinatorics, offering a quick way to calculate combinations. Recent discoveries have expanded its use, hinting at more secrets yet to be uncovered.

What is another name for Pascal's Triangle in Iran?

• Staircase of Mount Meru

• Blaise Triangle

• Yang Hui's Triangle

• Khayyam Triangle

How do you generate the next row in Pascal's Triangle?

• Multiply the numbers in the row

• Add the numbers in pairs

• Subtract the numbers in pairs

• Divide the numbers in pairs

What do the coefficients in Pascal's Triangle represent in a binomial expansion?

• The sum of the variables

• The product of the variables

• The exponents of the variables

• The coefficients of the variables

What pattern do you get when you add up the numbers in each row of Pascal's Triangle?

• Successive powers of two

• Successive powers of three

• Prime numbers

• Fibonacci sequence

What is the result of treating each number in row two as part of a decimal expansion?

• 100

• 121

• 144

• 169

What are the numbers in the third diagonal of Pascal's Triangle called?

• Square numbers

• Triangular numbers

• Pentagonal numbers

• Hexagonal numbers

What fractal pattern emerges when you shade all the odd numbers in Pascal's Triangle?

• Mandelbrot set

• Cantor set

• Koch snowflake

• Sierpinski's Triangle

In the context of Pascal's Triangle, what does the binomial expansion (girl + boy)^5 represent?

• The probability of having five boys

• The probability of having five girls

• The probability of having three girls and two boys

• The probability of having two girls and three boys

How many possible groups of five can be chosen from a group of twelve friends?

• 252

• 462

• 792

• 132

What recent discovery have mathematicians made about Pascal's Triangle?

• A way to expand it to new polynomials

• A new geometric application

• A new pattern in the triangle

• A new way to generate it