A. How many ways can the letters of the word ALGORITHM be arranged in a row? b. How many ways can the letters of the word ALGORITHM be arranged in a row if A and L must remain together (in order) as a unit?
c. How many ways can the letters of the word ALGORITHM be arranged in a row if the letters GOR must remain together (in order) as a unit?

a) 362880   b) 40320  c)720

Step-by-step explanation:

a) There are  9 letters in the word  ALGORITHM. Pls note that all 9 letters are different.

So to find the number of the ways which the letters can be arranged lets imagine , that in 1st place can stay any of 9 letters, in second place any of remaining 8 letters, in 3-rd place - any of 7 letters etc

So total number of the ways is

N= 9*8*7*6*5*4*3*2*1=9!=362880

b) If A and L must remain togetherin order AL ( LA is forbidden !) so we can imagine that AL is 1 special letter.

So total number of the letters is 8. Similarly to a) we can find the number of the ways as:

N=8*7*6*5*4*3*2*1=40320

c) Similarly to b) lets imagine that GOR is 1 special letter. So there are only 7 letters . Similarly to a) the number of the ways is

N=7*6*5*4*3*2=720

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,  

  b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,  

  and solve for the variable

* lets solve the problem

∵

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴

- use the rule

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴

* the solution of the equation is 2.23 to the nearest 2 decimal places

y = -1

x = 8

hope this !