Question

1. 5 People are attending a concert and have reserved 5 seats all in the same row. Persons 1 & 2 are a couple and want to sit together. How many ways can these people arrange themselves in the seats?

   1. a) Using this information and Excel, create a table that depicts all possible arrangements of the seats.

   2. b) Create a probability model to calculate the number of arrangements using Excel function. Hints: In creating your probability model, you will use the Product Rule and Permutations counting rules. There are more than 40 and less than 75 arrangements.

   For 1 bonus point, calculate the probability that person 1 and 3 are seated next to each other using Excel’s Count functions and your table. Still assume that Persons 1 & 2 are sitting next to each other.

Answer 1

Solution

Given that,

5 People are attending a concert and have reserved 5 seats all in the same row. Persons 1 & 2 are a couple and want to sit together.

How many ways can these people arrange themselves in the seats:

(a).

Using this information and Excel, create a table that depicts all possible arrangements of the seats:

Let the people are "a", "b", "c", ''d", and "e" where a and b are a couple who want to sit together.

Since a and b are always sitting together and total 5 sits are available hence the number of ways these 5 people can sit in,

=(2 4!)

=2 4 3 2 1

=48
The table:

Arrangement No	Arrangement
1	abcde
2	abced
3	abedc
4	abecd
5	abdce
6	abdec
7	bacde
8	baced
9	baedc
10	baecd
11	badce
12	badec
13	cabde
14	cabed
15	eabdc
16	eabcd
17	dabce
18	dabec
19	cbade
20	cbaed
21	ebadc
22	ebacd
23	dbace
24	dbaec
25	cdabe
26	dcabe
27	ecabd
28	ceabd
29	edabc
30	deabc
31	cdbae
32	dcbae
33	ecbad
34	cebad
35	edbac
36	debac
37	cdeab
38	cedab
39	dceab
40	decab
41	ecdab
42	edcab
43	cdeba
44	cedba
45	dceba
46	decba
47	ecdba
48	edcba

(b).

Create a probability model to calculate the number of arrangements using an Excel function.:

Given,

Hints:

In creating your probability model, you will use the Product Rule and Permutations counting rules.

There are more than 40 and less than 75 arrangements.

Since a and b are always sitting together let's assume and b as single. Hence we have 4 persons and 4 seats where this number of people can be arranged in 4!=4321=24 different ways.

Now a and b can interchange their positions also. So due to their interchange of seats, the different ways these 5 people can sit is 2 24=48

Now the total sample points 5!= 54321=120

Hence the corresponding probability is =0.4.