1.
Find a collection of identical objects, perhaps pennies or sugar cubes. Also grab a few dividers, which could be toothpicks, matches, or pens.
Line up the pennies in a single row. We will divide the row into some number of groups by placing the toothpicks in the spaces between the pennies. We will distinguish between which order the groups come in. For example, two different ways to divide 7 pennies into 4 groups would look like this:
We will allow for one or more groups to contain no pennies (by having two toothpicks next to each other or before or after all of the pennies).
By lining up the correct number of pennies and sticks, count the number of ways you can divide a row of pennies into the given number of groups.
(a)
If you want to divide a row of pennies into 4 groups, how many toothpicks will you need?
(b)
How many ways are there to separate a row of three pennies into two groups?
(c)
How many ways are there to separate a row of four pennies into two groups?
(d)
How many ways are there to separate a row of five pennies into two groups?
(e)
How many ways are there to separate a row of three pennies into three groups?
(f)
How many ways are there to separate a row of four pennies into three groups?
(g)
How many ways are there to separate a row of five pennies into three groups?
(h)
Based on your answers above, make a conjecture about how many ways you could separate a row of seven pennies into four groups.
Hint.
Look for the numbers you found in the previous questions in Pascal’s triangle
