Respuesta :
The product of a whole number 'n' with every whole number until 1 is called the factorial. There are 362880 ways to arrange the toys when there are no restrictions.
What is a factorial?
The product of a whole number 'n' with every whole number until 1 is called the factorial. The factorial of 4 is, for example, 43221, which equals 24.
Given that there are Nine stuffed animals, all different. Therefore, the number of ways for the different cases can be written as shown below.
a) There are no restrictions.
Given no constraints, the number of ways to arrange 9 toys is calculated as the factorial of 9 things.
The number of ways = 9!
Therefore, there are 9! = 362880 ways to arrange the toys here.
b) Cookie Monster must be in the middle.
The factorial of 8 is the number of different configurations in which Cookie Monster is in the centre.
The number of ways = 8!
This is because of the position of Cookie Monster.
Hence, there are 8! = 40320 ways to arrange the toys here.
c) BERT and ERNIE must be together.
The number of ways to arrange BERT and ERNIE so that they must be together is calculated as follows:
Number of ways
= The number of factorial of 8 elements, one of which is BERT and EARNIE combined × Number of factorial of BERT and EARNIE
= 8! × 2!
= 80640
Hence, there are 80,640 ways to arrange toys so that BERT and ERNIE are together
d) BERT, ERNIE, and OSCAR must be together.
The number of ways to arrange BERT, ERNIE, and OSCAR so that they must be together is calculated as follows:
Number of ways
= The number of factorial of 7 elements, one of which is BERT, EARNIE and OSCAR combined × Number of factorial of BERT, EARNIE and OSCAR
= 8! × 3!
= 241,920
Hence, there are 341,920 ways to arrange toys so that BERT, EARNIE and OSCAR are together.
e) BERT, ERNIE, and OSCAR must not be together.
Arrange the remaining 5 toys in 5! different ways first, and then we have 6 locations for the three toys that cannot be together. As a result, the number of ways to arrange the remaining three is a permutation of six elements taken three at a time.
Number of ways = 5! × 6 × 5 × 4
= 14400
Hence, there are 14400 ways to arrange it here.
f) They must not be in alphabetical order.
Given that they cannot be in alphabetical order, which can only occur in one way. Therefore, the total number of configurations conceivable here is calculated as:
Number of ways = 9! - 1
= 362879
Hence, there are 362879 ways to arrange the toys.
Learn more about Factorial:
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