Why doesn't the answer take into account overcounting of combinations?

[member 428835]

hey all

if i have, say 10 y's, 5 x's, and 4 z's and i want to see how many groups of 6 i can make with at least 5 y's, i know the answer is {10 \choose 5}{9 \choose 1}+{10 \choose 6} which is the combinations of 5 x's added to the combinations of 6 x's.

but why isn't the answer {10 \choose 5}{14 \choose 1}, which is the 5 x's and then the remaining 9 non-x characters added to the last 5 x-characters?

please help!

thanks!

josh

 

[mathman]

Your statement has many typos making it hard to understand. x's and y's are mixed up.

 

[member 428835]

sorry about this

this should make more sense:

if i have, say 10 y's, 5 x's, and 4 z's and i want to see how many groups of 6 i can make with at least 5 y's, i know the answer is {10 \choose 5}{9 \choose 1}+{10 \choose 6} which is the combinations of 5 y's added to the combinations of 6 y's.

but why isn't the answer {10 \choose 5}{14 \choose 1}, which is the 5 y's and then the remaining 9 non-y characters added to the last 5 y-characters?

 

[Office_Shredder]

Your solution overcounts the ways that you could have picked 6 y's. Let's suppose the y's are numbered 1 through 10. Your solution says first if I pick five y's, say 1,2,3,4,5, then I can pick anyone of the remaining 14 characters and get a distinct solution. For example I could pick y number 6.

On the other hand, I could start with five y's, 2,3,4,5,6, and then out of the remaining 14 characters I could pick y number 1, and end up with the exact same set of y's as above. Your attempt counts these as two separate ways of picking 1,2,3,4,5,6 out of my set of y's.