Determine the total number of arrangements of three or four toys from a basket of 8 different toys. Show your calculations
8P3 + 8P4 = ?
Thank you so much we lack common sense
Well, let's dive into the world of toy arrangements, shall we? To determine the total number of arrangements, we need to consider two cases: arranging three toys and arranging four toys. Let's tackle them one by one:
Case 1: Arranging three toys:
To calculate this, we'll use the concept of combinations. We want to choose 3 toys out of the 8 available. So, the number of arrangements of three toys can be calculated as follows:
Total number of arrangements = 8C3 = (8!)/(3!(8-3)!) = (8!)/(3!5!) = (8*7*6)/(3*2*1) = 56
Case 2: Arranging four toys:
Just like before, we'll use combinations to determine the total number of arrangements. Here we want to choose 4 toys out of the 8 available:
Total number of arrangements = 8C4 = (8!)/(4!(8-4)!) = (8!)/(4!4!) = (8*7*6*5)/(4*3*2*1) = 70
So, in total, there are 56 possible arrangements of three toys and 70 possible arrangements of four toys.
Now that we're done with the calculations, I hope these toy arrangements bring a smile to your face! 😄
To determine the total number of arrangements of three or four toys from a basket of 8 different toys, we need to consider both cases separately.
Case 1: Arrangements of three toys
To find the number of arrangements of three toys, we can use the combination formula. The formula for combinations is given by:
C(n, r) = n! / (r!(n - r)!)
In this case, we have n = 8 (total number of toys) and r = 3 (number of toys to be chosen). Substituting the values, we have:
C(8, 3) = 8! / (3!(8 - 3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
So, there are 56 different arrangements of three toys from a basket of 8.
Case 2: Arrangements of four toys
Similarly, to find the number of arrangements of four toys, we use the combination formula:
C(n, r) = n! / (r!(n - r)!)
Here, n = 8 (total number of toys) and r = 4 (number of toys to be chosen). Substituting the values, we have:
C(8, 4) = 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 from a basket of 8.
Therefore, the total number of arrangements of three or four toys from a basket of 8 different toys is 56 + 70 = 126.
To determine the total number of arrangements of three or four toys from a basket of 8 different toys, we can use the concept of combinations.
For arrangements of three toys, we use the formula for combinations. The number of combinations of choosing r objects from a set of n objects is given by the formula:
C(n, r) = n! / (r! * (n-r)!)
Plugging in the values for our case, where we have 8 toys and want to choose 3 toys, we calculate:
C(8, 3) = 8! / (3! * (8-3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
Therefore, there are 56 different arrangements of three toys from the basket of 8 toys.
For arrangements of four toys, we will use the same formula:
C(8, 4) = 8! / (4! * (8-4)!)
= 8! / (4! * 4!)
= (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1)
= 70
Hence, there are 70 different arrangements of four toys from the basket of 8 toys.