Without multiplying all the terms, verify these equalities. a) 10! = 6!7! b) 10!=7!5!3! c) 16! = 14!5!2! d) 9!=7!3!3!2!

a) 10! = 6*5*4*3*2 * 7!

10! = 6! 7! (Verified)

b) 10! = 3! * 5! * 7! = 7!5!3! (Verified)

c) 16! = 2! * 5! * 14!

16! = 14!5!2! (Verified)

d) 9! = 3! * 3! * 2! * 7!

9! = 7!3!3!2! (Verified)

Step-by-step explanation:

a) 10! = 6!7!

10! = 10*9*8*7!

Reducing to their lowest factors

10! = (2*5) * (3*3) * (2*2*2) * 7!

Rearranging the factors, we have;

10! = (2*3) * (5) * (2*2) * (3) * 2 * 7!

10! = 6*5*4*3*2 * 7!

10! = 6! 7! (Verified)

b) 10!=7!5!3!

10! = 10*9*8*7!

10! = (2*5) * (3*3) * (2*2*2) * 7!

Rearranging the factors, we have;

10! = (2*3) * (5) * (2*2) * (3) * 2 * 7!

10! = (3*2*1) * (5*4*3*2*1) * 7!

10! = 3! * 5! * 7! = 7!5!3! (Verified)

c) 16! = 14!5!2!

16! = 16*15*14!

16! = (2*2*2*2) * (5*3) * 14!

Rearranging the factors.

16! = (2) * (5) * (2*2) * (3) * (2) * 14!

16! = (2!) * (5*4*3*2) * 14!

16! = 2! * 5! * 14!

16! = 14!5!2! (Verified)

d) 9!=7!3!3!2

9! = 9*8*7!

9! = (3*3) * (2*2*2) * 7!

Rearranging the factors

9! = (3*2) * (3*2) * (2) * 7!

9! = 3! * 3! * 2! * 7!

9! = 7!3!3!2! (Verified)