[Math] what is the difference between ${10 \choose 2}$ and ${10 \choose 1}\times{9 \choose 1}$

combinations

I'm not able to understand the difference between these two.
Don't both just give the number of ways of selecting $2$ objects from a total of $10$?

Maybe the difference is that ${10 \choose 2}$ gives the number of ways of selecting $2$ distinct objects from a total of $10$, but ${10 \choose 1}$ gives the number of ways of selecting $1$ object from a total of $10$, but ${9 \choose 1}$ gives the number of ways of selecting $1$ object from a possible $9$ left options, so shouldn't multiplying ${10 \choose 1}$ with ${9 \choose 1}$ give the same result logicaly?

Best Answer

The difference is the duplication of pairs you get from 10C1 followed by 9C1.

For example, if the set you are choosing from is {a, b, c, d, e, f, g, h, i, j}

Then the choices for 10C1 followed by 9C1 will include (a, b) and (b, a) whereas the choices for 10C2 will include only one of the two.

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