All of them please, fast and clear A classroom has two rows of eight seats each.
ID: 3854036 • Letter: A
Question
All of them please, fast and clear 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_i elementof Z, for 1 lessthanorequalto i lessthanorequalto n. If a divides each c_i, then a|(c_1 x_1 + c_2 x_2 + .. + c_n x_n) where x_i elementof Z for all 1 lessthanorequalto i lesthanorequalto 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 the logical statement? What is the symmetric difference of two sets? What is the principle of mathematical induction.Explanation / Answer
1)Solution:
->In the front row, there are eight empty seats, 5 of which will be occupied by front row only students. Thus we have (after permuting these students’ positions in the selected seats)
p(8,5)=6720 ways that these same five students can sit in the front.
->Likewise, in the back, there are also eight empty seats, 4 or which will be occupied by back row only students. Thus we have (after permuting these students’ positions in the selected seats)
p(8,4)=1680 separate ways that the same four students can sit in the back.
->Of the remaining seven seats in either rows, only five will be occupied by the remaining free floating students, thus we have (after permuting these students’ positions in the selected seats)
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)=28449792000
Then the total number of seating arrangements is: = 28449792000