A classroom has two rows of eight seats each. There are 14 students, 5 of whom a

Question

A classroom has two rows of eight seats each. There are 14 students, 5 of whom always sit in the front row and 4 of whom always sit in the back row. In how many ways can the students be seated? Show the validity of the following argument Prove that: Let c_1 elementof Z, for 1 lessthanorequalto i lessthanorequalto n. If a divides each c_1, then a|(c_1x_1 + c_2x_2 + ellipsis c_nx_n) where x_1 elementof Z for all 1 lessthanorequalto i lessthanorequalto n. Define the greatest common divisor and the least common multiple of two integers. What is the relation between the greatest common divisor and the least common multiple? What is a logical statement? What is the symmetric difference of two sets? What is the principle of mathematical induction? When we can use this principle? Prove the principle of mathematical induction.

Explanation / Answer

1)Solution:

We will regard the eight seats in each row different. The arrangement process can be divided into three stages.

1. Let the 5 students who always sit in the font row choose their seats. There are P(8, 5) possible outcomes.

2. Let the 4 students who always sit in the back row choose their seats. There are P(8, 4) possible outcomes.

3. Let the remaining five students to fill choose their seats from the remaining 16 9 seats. There are P(7, 5) different ways.

By the multiplication principle, the total number of ways to seat the students is P(8, 5)·P(8, 4)· P(7, 5).

P(8, 5)·P(8, 4)· P(7, 5)=28449792000

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