Question

the organizer of a conference is selecting workshops to include. she will select from 5 workshops about anthropology and 8 workshops about psychology. in how many ways can she select 6 workshops if fewer than 3 must be about anthropology?

Answer 1

Given data: 5 workshops from anthropology

8 workshops from psychology

Since we are to select from the workshops, this is a combination question

$^pC_q=\frac{p!}{(p-q)!\times q!}$

If from 6 workshops selected, she has to select fewer than 3 from anthropology

This means that there are 3 possible scenarios

case 1:

select 2 from the anthropology and 4 from the psychology workshop

$^5C_2\times^8C_4=10\times70=700\text{ ways}$

case 2:

select 1 from the anthropology and 5 from the psychology workshop

$^5C_1\times^8C_5=5\times56=280\text{ ways}$

case 3: select 0 from the anthropology and 6 from the psychology workshop

$^5C_0\times^8C_6=1\times28=28\text{ ways}$

Therefore the total number of ways will be the sum of all the ways in the cases

= 700+280+28 ways

= 1008 ways

Therefore there are 1008 ways