Darlene is adopting a pet. she can choose from 8 kittens and 6 puppies.How many ways can she choose a pet to adopt

Mathematics

2 answers:

juin [17]3 years ago

7 0

14 cause shes only getting 1 pet and there are 14 animals in total

Mandarinka [93]3 years ago

6 0

Answer: The required number of ways is 14.

Step-by-step explanation: Given that Darlene is adopting a pet and she can choose from 8 kittens and 6 puppies.

We are to find the number of ways in which she can choose a pet to adopt.

We have

the total number of pets that are available to choose from is given by

$n=8+6\\\\\Rightarrow n=14.$

Since Darlene is going to adopt only one pet, so we have

$r=1.$

Therefore, the number of ways in which she can choose a pet to adopt is given by

$^nC_r\\\\=^{14}C_1\\\\\\=\dfrac{14!}{1!(14-1)!}\\\\\\=\dfrac{14!}{1!13!}\\\\\\=\dfrac{14\times13!}{1\times13!}\\\\=14.$

Thus, the required number of ways is 14.

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Eights rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other r

erastova [34]

Answer:

The probability is $\frac{56!}{64!}$

Step-by-step explanation:

We can divide the amount of favourable cases by the total amount of cases.

The total amount of cases is the total amount of ways to put 8 rooks on a chessboard. Since a chessboard has 64 squares, this number is the combinatorial number of 64 with 8, $64 \choose 8 .$

For a favourable case, you need one rook on each column, and for each column the correspondent rook should be in a diferent row than the rest of the rooks. A favourable case can be represented by a bijective function $f : A \rightarrow A ,$ with A = {1,2,3,4,5,6,7,8}. f(i) = j represents that the rook located in the column i is located in the row j.

Thus, the total of favourable cases is equal to the total amount of bijective functions between a set of 8 elements. This amount is 8!, because we have 8 possibilities for the first column, 7 for the second one, 6 on the third one, and so on.

We can conclude that the probability for 8 rooks not being able to capture themselves is

$\frac{8!}{64 \choose 8} = \frac{8!}{\frac{64!}{8!56!}} = \frac{56!}{64!}$