Question

Help Health food magazine has 14 advertising sales people. 4 men and 10 women. 6 salespeople will be randomly selected to attend the annual health food convention in Las Vegas.
What is the probability at of at least 3 men being selected.

Answer 1

`at least 3 men` means 3 or more men.


There are 4 men total, so we could have 3 men or 4 men selected. There are no other possible options.


So what I would do is break up the problem into 2 cases


Case A will be where 3 men are selected. Case B will be where 4 men are selected.


-----------------------------------------

Case A) 3 men are selected

Use the combination formula to determine how many ways we can pick 3 men
\[\Large _n C _r = \frac{n!}{r!*(n-r)!}\]
\[\Large _4 C _3 = \frac{4!}{3!*(4-3)!}\]
\[\Large _4 C _3 = \frac{4!}{3!*1!}\]
\[\Large _4 C _3 = \frac{4*3*2*1}{3*2*1*1}\]
\[\Large _4 C _3 = \frac{24}{6}\]
\[\Large _4 C _3 = 4\]
There are 4 ways to select 3 men from a pool of 4.


Use the same formula to determine how many ways there are to pick 3 women
\[\Large _n C _r = \frac{n!}{r!*(n-r)!}\]
\[\Large _{10} C _3 = \frac{10!}{3!*(10-3)!}\]
\[\Large _{10} C _3 = \frac{10!}{3!*7!}\]
\[\Large _{10} C _3 = \frac{10*9*8*7*6*5*4*3*2*1}{3*2*1*7*6*5*4*3*2*1}\]
\[\Large _{10} C _3 = \frac{3,628,800}{30,240}\]
\[\Large _{10} C _3 = 120\]
There are 120 ways to select 3 women from a pool of 10


Multiply the previous results
4*120 = 480


So there are 480 ways to pick 3 men and 3 women (from a pool of 4 men and 10 women)


Hopefully this makes sense?

Answer 2

let me know if you're able to do case B @ray_sihota

Answer 3

than you it does make sense for case b I did 4c4 x10 c 2 and added that to 480 which give me 525/14 c 6 525/3003

Answer 4

You should find that
4 C 4 = 1
10 C 2 = 45
so 1*45 = 45

There are 45 ways to pick 4 men and 2 women (pool of 4 men and 10 women)

add this to the previous result from case A to get 480+45 = 525

so you have the correct values. Nice work

Answer 5

There are 525 ways to either pick 3 men or pick 4 men.

This value will be the numerator to be used later

Let's calculate the denominator

There are 4+10 = 14 people. There are 6 slots to fill. What is the value of 14 C 6 ?

Answer 6

oh you already wrote it out, you got 3003 which is correct

so the probability is 525/3003

Answer 7

make sure you reduce the fraction as much as possible.

In this case,
525/3003 = 25/143

so 25/143 is the final answer assuming your teacher wants a fraction as the answer. If s/he wants a decimal, then use a calculator to get the decimal form of 25/143