# New Question

• Dec 13th 2009, 08:16 AM
dayscott
choose x distinguishable devices from a set
another question concerning the binomial coefficient from my "discrete mathematics and its applications" book:

"there are 8 sorts of distinguishable devices. How many ways are there to choose a dozen devices with at least 3 of sort A and no more than 2 of sort B." solution: 9724

$\displaystyle {7-1+7 \choose 7} + {2-1+8 \choose 2} = 1716+36$

a dozen = 12. 12 - 3 ( at least 3 of sort A ) = 9. 9-2 = 7 - so i can choose 7 out of 7 sorts, this is the first part of the sum. the second part of the sum is my second choise of 2 out of 8 distinguishable devices. i sum them up because there is no intersection.

what did i do wrong :( ?
• Dec 13th 2009, 10:28 AM
Plato
Once again, the fastest way to this answer is by way of generating functions.
Find the coefficient of $\displaystyle x^{12}$ the expansion of $\displaystyle \left( {1 + x + x^2 } \right)\left( {\sum\limits_{k = 3}^{12} {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^6$.

But you seem not to be using generating functions in the class.

So here is tedious way.
$\displaystyle \binom{9+8-1}{9}$ is the number of ways to select a dozen including at least three of type A.

$\displaystyle \binom{6+8-1}{6}$ is the number of ways to select a dozen including at least three of type A and least three of type B.

Thus, $\displaystyle \binom{9+8-1}{6}-\binom{6+8-1}{6}=9724$
is the number of ways to select a dozen including at least three of type A but no more than two of type B.
• Dec 13th 2009, 10:48 AM
dayscott
Quote:

Originally Posted by Plato
Once again, the fastest way to this answer is by way of generating functions.
Find the coefficient of $\displaystyle x^{12}$ the expansion of $\displaystyle \left( {1 + x + x^2 } \right)\left( {\sum\limits_{k = 3}^{12} {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^6$.

But you seem not to be using generating functions in the class.

.

in class we had tasks like: $\displaystyle (x+y)^{25}$ whats the coefficient of $\displaystyle x^{13}$ ?

but .. i am not able to extrapolate in my brain how to use this in this excercise, if it is not to hard, may be you can try to explain?