How many ways are there to arrange 4 distinct math books and 5 distinct history books on a shelf if all of the math books have to be together, but the history books do not have to be together? a. 17, 280
b. 20
c. 2, 880
d. 1024
e. 24

The number of ways these books can be arranged is 17,280 ways

Step-by-step explanation:

number of ways 4 distinct math books can be arranged = 4! = 4 x 3 x 2 x 1 = 24 ways

number of ways of 5 distinct history book can be arranged = 5! = 5 x 4 x 3 x 2 x 1 = 120 ways

if all of the math books have to be together, but the history books do not have to be together,

↑ H₁ ↑ H₂ ↑ H₃ ↑ H₄ ↑ H₅ ↑ = 6 ways

The upward arrows represent, the number of ways all the maths books can be arranged with the 5 history books.

Total number of ways these books can be arranged = 24 x 120 x 6

                                                                                        = 17,280 ways

Therefore, the number of ways these books can be arranged is 17,280 ways

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).