Assuming that each letter can appear at most once in an arrangement, how many ways are there to arrange 10 letters taken from the alphabet a-z such that: (a) a is not included in the arrangement? (b) z is included in the arrangement? (c) both a and z are included in the arrangement? (d) exactly one of a and z appears in the arrangement? the intention here is that exactly one element from the set faz) appears in the string. so, abedetohit and bedestohij are ok, but abodedghi is not.(e) z and a appear and are adjacent to each other in the arrangement?

(a) 25C10 x 10! ways

(b) 25C9 x 10! ways

(c) 24C8 x 10! ways

(d) 2 x 24C9 x 10! ways

(e) 2 x 24C8 x 9! ways

Step-by-step explanation:

Given: We need to arrange 10 letters taken from the alphabet a-z such that:

(a) a is not included in the arrangement:

   As a is not included in arrangement, we need to select from remaining 25 alphabets. From 25 alphabets, 10 alphabets can be selected in 25C10 ways and can be arranged in 10! ways.

So, Total no. of ways = 25C10 x 10!

(b) z is included in the arrangement :

Now z is included in the arrangement, so we need to select 9 alphabets from 25 alphabets.  

From 25 alphabets, 9 alphabets can be selected in 25C9 ways and can be arranged in 10! ways.

So, Total no. of ways = 25C9 x 10!

(c) both a and z are included in the arrangement:

In this case, we need to select 8 alphabets from 24 alphabets.  

From 24 alphabets, 8 alphabets can be selected in 24C8 ways and can be arranged in 10! ways.

So, Total no. of ways = 24C8 x 10!

(d) Exactly one of a and z appears in the arrangement:  

In this case, we need to select one from a and z, so it can be done in 2C1 i.e 2 ways. Now, from 24 alphabets, 9 alphabets can be selected in 24C9 ways and can be arranged in 10! ways.

So, Total no. of ways = 2 x 24C9 x 10!

(e) z and a appear and are adjacent to each other in the arrangement :

In this case, we need to select 8 alphabets from 24 alphabets and this can be done in 24C8 ways. z and a can be arranged in 2 ways.

So, Total no. of ways = 2 x 24C8 x 9!

answer: 204

step-by-step explanation:

we are told that the revenue is

x⋅(4080−10x)=−10x^2+4080x

this is a quadratic function, so to maximize it we should find the x-value of its vertex. recall that the x-value of the vertex of the parabola ax^2+bx+c is given by the formula −b/2a.

so in this case, we have a=−10 and b=4080, so the x-value of the vertex is

−b/2a= −(4080)/2(−10)=204