A group of 10 students consists of 3 girls and 7 boys. How many ways can the students be
arranged from left to right if the first 3 students are girls and the remaining students are all
boys?
How many ways can the girls be arranged?
3!
how many ways can the boys be arranged?
7!
so
3!*7! big number
6*5040 = 30,240
no the girls can be arranged 9 different ways and the boys can be arranged 49 different ways
To determine the number of ways the students can be arranged, we need to consider the number of arrangements for the girls and the boys separately, and then multiply them together since the arrangements of the girls and boys are independent.
Since there are 3 girls in the first 3 positions, we only need to consider the arrangements for the remaining 7 boys.
The number of ways to arrange the 7 boys can be calculated using permutations, which is the number of ways to arrange objects without repetition.
In this case, we have 7 distinct objects (boys) to be arranged, and the order matters. Therefore, we can use the formula for permutations of n objects, which is n!.
So, the number of ways to arrange the 7 boys is 7!.
Now, let's calculate it:
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
Therefore, there are 5040 ways to arrange the 7 boys.
To find the total number of arrangements for all the students, we then multiply the number of ways to arrange the girls with the number of ways to arrange the boys.
Since there is only one way to arrange the 3 girls (since their positions are fixed), we multiply 1 with 5040.
1 x 5040 = 5040
So, there are 5040 ways to arrange the students from left to right, given the condition that the first 3 students are girls and the remaining students are all boys.