There are 5 boys and 4 girls in my class.
In how many ways can they be seated in a row of 9 chairs such that at least 2 boys are next to each other?
In how many ways can they be seated in a row of 9 chairs such that at least 3 girls are all next to each other?
look back at your other post
no reply
http://www.jiskha.com/display.cgi?id=1381013688
sorry, i mean you are correct?
but how about number 2. That was harder for me.
the question mark is a ! not a ?
To find the number of ways they can be seated in a row of 9 chairs, we can use the concept of permutations.
For the first question, let's first calculate the number of total ways to arrange all 9 students without any restrictions. Since there are 9 chairs, the total number of ways to seat the students is 9 factorial (9!).
Next, let's calculate the number of arrangements where no two boys are seated next to each other. We can treat the 5 boys as distinct, and we have 5! ways to arrange them among the 9 chairs. Similarly, we can treat the 4 girls as distinct, and we have 4! ways to arrange them.
However, we need to account for the fact that the boys and girls can still be seated in various arrangements as long as at least 2 boys are next to each other. From 2 boys seated next to each other, we can have groups of 2, 3, 4, or 5 consecutive boys. These groups can be seated in different ways, so we need to calculate the number of ways for each group and sum them up.
For example, if we have 2 consecutive boys, we treat them as a pair and have 4 more boys and 4 girls to arrange, resulting in (4!)(4!). Likewise, for 3 consecutive boys, we treat them as a group and have 3 more boys and 4 girls to arrange, resulting in (3!)(4!). The same logic applies to 4 and 5 consecutive boys.
Therefore, the total number of ways to seat the students such that at least 2 boys are next to each other is the difference between the total number of ways to seat all students and the number of ways to seat them with no two boys seated together. Mathematically, this is expressed as:
9! - [(5!)(4!) + (4!)(4!)(5) + (3!)(4!)(5) + (2!)(4!)(5)] = 9! - [(5!)(4!) + (4!)(4!)(5) + (3!)(4!)(5) + (2!)(4!)(5)]
For the second question, the logic is very similar. We just need to calculate the number of ways for a group of 3 girls seated consecutively and subtract it from the total number of ways to seat all students. Mathematically, this is expressed as:
9! - (3!)(4!)(7)
Therefore, to find the actual numerical values, you can compute the above expressions.