a Assuming that the order she puts on the bracelets does not matter, we want to know the number of combinations when picking 0, 1, 2, 3, and 4 bracelets from a group of 4.
Number of Bracelets
_nC_r
0
_4C_0
1
_4C_1
2
_4C_2
3
_4C_3
4
_4C_4
To calculate these expressions, we can use the graphing calculators built in combination function. First we enter the number of bracelets Donna has.
Next, push the MATH button, scroll to PRB, and choose the third option. Having chosen _nC_r, we finish by entering the number of bracelets she wants to pick out.
There is only one way she can pick no bracelets. We repeat the procedure with the remaining expressions.
As we can see, the number of possible combinations when she has four bracelets mirrors the fourth row of Pascal's triangle.
Alternative Solution
Calculate Without a Graphing Calculator
We can also calculate the number of combinations using the following formula.
_nC_r=_nP_r/r! ⇔ _nC_r=n!/(n-r)!r!
We will calculate one of the examples of combinations. Let's choose _4C_2.
b From Part A, we know that a given row in Pascal's triangle n mirrors the number of combinations when choosing r items from a group of n. Therefore, since we have 6 bracelets, we need to include the sixth row of Pascal's triangle.
We want to select 4 bracelets, which means we start counting from r=0 until we reach r=4 on this row.
As we can see, the number of combinations when choosing 4 of 6 bracelets is 15.
