Ginger has 12 plastic bead containers. The containers measure 1 inch on each side. How many rectangular prisms, each with a different-sized base, could Ginger make by stacking all of the bead containers?
different shapes:
1 1 12
1 2 6
1 3 4
2 2 3
number of different bases for each one:
1 1 12 --->1 1 12, 1 12, 1 ----- 2 of them
1 2 6 ---> 1 2 6, 1 6, 2, 2 6 1 --- 3 of them
1 3 4 ---> 1 3 4, 1 4 3, 3 4 1 --- 3 of them
2 2 3 ---> 2 2 3, 2 3 2 ----- 2 of them
10 different prisms
10 different prisms
how explain breifly
To find out how many rectangular prisms can be made by stacking all the bead containers, we need to determine the factors of 12, since the dimensions of each base must be different.
To find the factors of 12, we can list all possible combinations of two numbers that multiply to give 12:
1 x 12 = 12
2 x 6 = 12
3 x 4 = 12
So, the factors of 12 are: 1, 2, 3, 4, 6, and 12.
Now, we need to find the number of ways to select two distinct numbers from this set of factors. To do this, we use the combination formula, which is given by:
C(n, r) = n! / (r!(n - r)!)
In our case, n = 6 (the total number of factors) and r = 2 (since we want to choose two distinct numbers). Therefore, the number of ways to choose two distinct factors from the set of factors of 12 is:
C(6, 2) = 6! / (2!(6 - 2)!) = 6! / (2! * 4!) = (6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 4 * 3 * 2 * 1) = 15
Hence, Ginger can make 15 different rectangular prisms by stacking all the bead containers.