Aboy has 4 red , 5 yellow and 5 green marbles. in how many ways can the boy arrange the marbles in a line if: a. marbles of the same color are indistinguishable? b. all marbles have different sizes?

(a) since the same colored marbles are indistinguishable, then Red1 in the first place and Red2 in the seond place is the same as Red2 in the first place and Red1 in the second place. (which means: do not duplicate the count)

4! 5! 5! = 345,600

(b) since all of the marbles have different sizes, every combination counts.

14!  = 87, 178, 291, 200

which expression represents the approximate length of bc?

step-by-step explanation:

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."