How many empty seats are there?

Name: Ronnie
Who is asking: Parent
Level: Middle

Question:
Here is a homework problem that is killing me...please help!!

Three new students arrived in class. Teacher says there are 210 ways you can set. How many empty seats are there in the classroom? After new student are assigned seats, how many empty seats will remain?

Hi Ronnie,

Just to see how you would calculate this suppose that there were 4 empty seats. The first of the three students to arrive would have a choice of 4 seats. Whichever seat she chose the second student would have a choice of 3 seats. Thus the two students could seat themselves in 4 3 = 12 ways. Whatever seats they chose the third student could choose either of the 2 remaining seats so altogether the three students could seat themselves in

2 = 24 ways Hence 4 empty seats is much too small. Lets try 10 empty seats.

The first of the three students to arrive would have a choice of 10 seats. Whichever seat she chose the second student would have a choice of 9 seats. Thus the two students could seat themselves in 10 9 = 90 ways. Whatever seats they chose the third student could choose any of the 8 remaining seats so altogether the three students could seat themselves in

8 = 720 ways Thus 10 empty seats is too large.

Try something between 4 and 10.

The method of "guess and check" is a perfectably reasonable way to solve this problem. At this point however you might see the pattern. If there are n empty seats then the number of ways that the three students can seat themselves is

n(n - 1)(n - 2) Hence you want n so that n(n - 1)(n - 2) = 210 Writing 210 in terms of its prime factors I get 210 = 7

2 Thus n(n - 1)(n - 2) = 7

2 Can you see what n has to be?

Penny

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