gokyada wrote:Can someone clearly explain why its 7C1 * 10C2 and not 7C1 plus 10C2? Am missing some core concept here
Hi,
here's the rule known as the Fundamental Counting Principle:
If there are m ways to make one selection, and n ways to make another selection, then as long as those selections are independent, there are m*n ways to make both selections.
Here's another way to remember it:
whenever you make MULTIPLE selections, MULTIPLY; and
whenever you make ALTERNATIVE selections, ADD.
Since we're selecting one person for the first position AND 2 people for the second pair of positions, we MULTIPLY.
Words such as "and", "both" and "all" indicate multiple selections; words such as "or", "at least" and "at most" indicate alternative selections.
Here's a simple example to help us understand the rule.
A class contains 3 girls and 2 boys. If one girl and one boy are going to be selected to dance together, how many different pairs of dancers can be formed?
Let's call the girls A, B and C and the boys F and G. We're choosing 1 out of 3 girls, so there are 3 different selections we can make; we're choosing 1 out of 2 boys, so there are 2 different selections we can make. We're choosing a girl AND a boy, so we multiply:
3*2 = 6
If we were to actually count the pairs, we'd confirm our calculation:
AF
AG
BF
BG
CF
CG
As you can see from the list, for each girl we select, there are two possible boys - that's why we multiply.
Now let's change the question a bit:
A class contains 3 girls and 2 boys. If one girl or one boy is going to be chosen to be class president, how many different presidents can be chosen?
Keeping the same names, we see that out of the girls, we have 3 different choices; out of the boys, we have 2 different choices. Since we're selecting 1 girl OR 1 boy, we ADD the results: 3+2=5.
Again, if we were to list them out, we confirm our math:
A
B
C
D
E
As you can see from this list, there's no combining at all, so we simply add the results together.
Going back to the original question, we have 7 possible choices for the first position and 45 for the second pair; since we're choosing 1 of 7 AND 2 of 10, we multiply to get the final answer. If we were to actually list it out (calling the first group A through F and the second group G through P):
AGH
AGI
AGJ
.
.
.
we quickly see that there are a LOT more than 17 possibilities.
