My friend lost 2 charms off her 7-charm bracelet. for her birthday, i bought her a new charm to replace one of the lost ones. unfortunately, i messed up and got her a duplicate of one of the charms she still has. how many distinguishable ways can she put her 6 charms on her bracelet? (two of the charms are the same, rotations are indistinguishable, and turning the bracelet front-to-back is indistinguishable.)

30 ways

Step-by-step explanation:

Given

there are total of 6 charms out of which 2 are same

total no of ways in which n person can be arranged in a round table is (n-1)!

So 6 charms can be arranged in a bracelet is (6-1)!

Now two charms are same i.e. total no of ways to arrange

As rotation are indistinguishable i.e. clockwise become anti-clockwise therefore

total no of ways are                    

the answer is (-8,-5)

step-by-step explanation:

the endpoint to midpoint is half the length. the length is (-2,-1) so when you are at the midpoint just add it to it to find your answer -6+(-2)= -8 and -4+(-1)=-5 so (-8,-5)