1. Determine the total number of
arrangements of three or four toys from a
basket of eight different toys.
To determine the total number of arrangements of three or four toys from a basket of eight different toys, we can use the concept of combinations.
To find the total number of arrangements of three toys, we will use the combination formula:
nCr = n! / (r!(n-r)!)
Here, n represents the total number of toys in the basket (8), and r represents the number of toys we want to select (3).
So, substituting the values into the formula:
8C3 = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Therefore, there are 56 different arrangements of three toys.
To find the total number of arrangements of four toys, we will again use the combination formula, this time with n = 8 and r = 4:
8C4 = 8! / (4!(8-4)!) = 8! / (4!4!) = (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1) = 70
So, there are 70 different arrangements of four toys.
Therefore, the total number of arrangements of three or four toys from a basket of eight different toys is 56 + 70 = 126.