At a birthday party, 5 boys and 3 girls are seated around a table. How many different arrangements are possible if
b
a boy named Alex, and a girl named Beth are not seated next to each other?
Well, let's see... We have 5 boys and 3 girls, so there are a total of 8 people. If we ignore the restriction about Alex and Beth, there would be 8! (factorial) possible arrangements. However, we need to eliminate the arrangements where Alex and Beth are seated next to each other.
Now, think about it this way. If we treat Alex and Beth as a single entity, there would be 7 entities (6 single boys and the "AlexBeth" pair) to arrange around the table. There are 7! possible arrangements when Alex and Beth are seated next to each other.
But we don't want that. So, we need to consider the cases where Alex and Beth are NOT seated next to each other. There are two ways for that to happen - either Alex is on the left of Beth, or he is on the right. So, for each of these cases, there would be 7! arrangements.
Therefore, the total number of different arrangements where Alex and Beth are not seated next to each other will be 2 * 7!.
And if you want me to calculate the exact number, I'm afraid I don't have enough clown fingers to do the math for you. Maybe you could give it a try?
To find the number of different arrangements where a boy named Alex and a girl named Beth are not seated next to each other, we can use the principle of complementary counting.
First, let's calculate the total number of arrangements without any restrictions:
The total number of people (boys + girls) at the party is 5 + 3 = 8.
The total number of ways to arrange 8 people around a circular table is (8-1)! = 7!.
Now, let's calculate the number of arrangements where Alex and Beth are seated next to each other:
We can consider Alex and Beth as a single entity since they must be seated together. This means we need to arrange 6 entities around the table: {Alex&Beth, Boy1, Boy2, Boy3, Boy4, Girl1, Girl2}.
The number of ways to arrange 6 entities around a circular table is (6-1)! = 5!.
Now, we need to remember that Alex and Beth can be arranged within their pair in 2 different ways (Alex followed by Beth or Beth followed by Alex).
Therefore, the number of arrangements where Alex and Beth are seated next to each other is 5! * 2.
Finally, we can calculate the number of arrangements where Alex and Beth are not seated next to each other by subtracting the previous result from the total number of arrangements:
Number of arrangements = Total arrangements - Arrangements with Alex and Beth seated next to each other
= 7! - 5! * 2
= 5040 - 120 * 2
= 5040 - 240
= 4800.
So, there are 4800 different arrangements possible if a boy named Alex and a girl named Beth are not seated next to each other at the birthday party.
To solve this problem, we can use the concept of counting the complement or the total number of arrangements and subtracting the arrangements where Alex and Beth are seated next to each other.
First, let's find the total number of arrangements without any restrictions. We have 8 people to seat around a circular table, so we'll start by seating the person at the head of the table. There are 8 choices for the first person.
After that, we'll seat the other 7 people one by one. For the second person, there are 7 choices remaining since one seat is already occupied. For the third person, there are 6 choices left, and so on.
Therefore, the total number of arrangements without any restrictions is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320.
Now, let's find the number of arrangements where Alex and Beth are seated next to each other. Since Alex and Beth must be seated together, we can treat them as a single unit. Now we have 7 units to arrange around the table (including the "Alex-Beth" unit), which can be done in (7-1)! = 6! = 720 ways.
However, within the "Alex-Beth" unit, there are 2 possible arrangements: Alex can be seated on the left side of Beth or on the right side. So we need to multiply our previous result (720) by 2.
Therefore, the number of arrangements where Alex and Beth are seated next to each other is 720 × 2 = 1,440.
Finally, to find the number of arrangements where Alex and Beth are not seated next to each other, we subtract the number of arrangements where they are seated next to each other from the total number of arrangements:
Number of arrangements without Alex and Beth together = Total number of arrangements - Number of arrangements with Alex and Beth together
= 40,320 - 1,440
= 38,880.
So, there are 38,880 different arrangements possible where Alex and Beth are not seated next to each other at the birthday party.