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Vika [28.1K]
3 years ago
11

Factorize 9(a+1)^2-81​

Mathematics
1 answer:
Lunna [17]3 years ago
5 0

Answer:9a^2+18a-72

Step-by-step explanation:

9(a+1)^2-81

9(a+1)(a+1)-81

Open brackets

9(a^2+a+a+1)-81

9(a^2+2a+1)-81

9a^2+18a+9-81

9a^2+18a-72

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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
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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!}

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