Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

There is a deck with 60 cards, with 11 different types of cards. It contains 20 A cards, 4 B cards, 4 C cards, 4 D cards, 4 E cards, 4 F card, 4 G cards, 4 H cards, 4 I cards, 4 J cards and 4 K cards ( for a total of 60 cards).

We pick 7 cards to our starting hand. How many different different starting hands are there? The order of the cards in the hand does not matter.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

It's the number of solutions of $$a_0+a_1+\cdots+a_{10}=7,\quad0\le a_0\le7,\quad0\le a_i\le4{\rm\ for\ }1\le i\le10$$ This in turn is the coefficient of $x^7$ in $$(1+x+\cdots+x^7)(1+x+x^2+x^3+x^4)^{10}$$ which is $$(1-x^8)(1-x^5)^{10}(1-x)^{-11}$$ Ignoring terms $x^8$ and higher, this is $$(1-10x^5)(1+11x+66x^2+\cdots)$$ where the coefficient of $x^n$ in the second bracket is $10+n\choose n$. So it seems the answer is $${17\choose7}-660$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.