A pharmaceutical company interviews 12 people for eight openings in the research department of the company. 7 of the 12 people are women. If all 12 are qualified, in how many ways can the employer fill the 8 positions if (a) the selection is random

(b) exactly three selections are men?

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(a) 495, (b) 210

Step-by-step explanation:

Total number of persons = 12

number of openings = 8

number of women = 7

number of men = 12 - 7 = 5

(a) Number of ways to select the 8 persons

(b) Three are men

So, 3 men and 5 women

The number of ways are

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

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