The comments were, in my opinion, closer the first time -- this is a standard application of inclusion-exclusion and could be a homework exercise. The number of ways to choose $k$ boxes from a set of $i$ rows (without worrying about hitting all the rows) is $\binom{in}{k}$. The number of ways to choose $i$ rows is $\binom{n}{i}$. Thus the total number of ways to make your choices so that every row gets hit is $$\sum_{i = 0}^n (-1)^{n - i} \binom{in}{k} \binom{n}{i}.$$ (Perhaps this can be rewritten in other ways that appeal to you more.)

A suggestion to Sheldon: this question has a very elementary look and feel (and indeed it turns out to be very elementary). When posting a problem of this sort, you should provide some information about why it's of interest to you, so as to help distinguish the homework-cheaters from people who have interesting (e.g., research-related) reasons for needing the solution to elementary problems. I also agree with Gerry's final comment and an-mo-user's comment here: http://tea.mathoverflow.net/discussion/947/ballsandbins-type-problem-question-closed

The comments were, in my opinion, closer the first time -- this is a standard application of inclusion-exclusion and could be a homework exercise. The number of ways to choose $k$ boxes from a set of $i$ rows (without worrying about hitting all the rows) is $\binom{in}{k}$. The number of ways to choose $i$ rows is $\binom{n}{i}$. Thus the total number of ways to make your choices so that every row gets hit is $$\sum_{i = 0}^n (-1)^{n - i} \binom{in}{k} \binom{n}{i}.$$ (Perhaps this can be rewritten in other ways that appeal to you more.)

A suggestion to Sheldon: this question has a very elementary look and feel (and indeed it turns out to be very elementary). When posting a problem of this sort, you should provide some information about why it's of interest to you, so as to help distinguish the homework-cheaters from people who have interesting (e.g., research-related) reasons for needing the solution to elementary problems. I also agree with Gerry's final comment and an-mo-user's comment here: http://mathoverflow.tqft.net/discussion/947/ballsandbins-type-problem-question-closed

The comments were, in my opinion, closer the first time -- this is a standard application of inclusion-exclusion and could be a homework exercise. The number of ways to choose $k$ boxes from a set of $i$ rows (without worrying about hitting all the rows) is $\binom{in}{k}$. The number of ways to choose $i$ rows is $\binom{n}{i}$. Thus the total number of ways to make your choices so that every row gets hit is $$\sum_{i = 0}^n (-1)^i \binom{in}{k} \binom{n}{i}.$$ (Perhaps this can be rewritten in other ways that appeal to you more.)

A suggestion to Sheldon: this question has a very elementary look and feel (and indeed it turns out to be very elementary). When posting a problem of this sort, you should provide some information about why it's of interest to you, so as to help distinguish the homework-cheaters from people who have interesting (e.g., research-related) reasons for needing the solution to elementary problems. I also agree with Gerry's final comment.

The comments were, in my opinion, closer the first time -- this is a standard application of inclusion-exclusion and could be a homework exercise. The number of ways to choose $k$ boxes from a set of $i$ rows (without worrying about hitting all the rows) is $\binom{in}{k}$. The number of ways to choose $i$ rows is $\binom{n}{i}$. Thus the total number of ways to make your choices so that every row gets hit is $$\sum_{i = 0}^n (-1)^{n - i} \binom{in}{k} \binom{n}{i}.$$ (Perhaps this can be rewritten in other ways that appeal to you more.)

A suggestion to Sheldon: this question has a very elementary look and feel (and indeed it turns out to be very elementary). When posting a problem of this sort, you should provide some information about why it's of interest to you, so as to help distinguish the homework-cheaters from people who have interesting (e.g., research-related) reasons for needing the solution to elementary problems. I also agree with Gerry's final comment and an-mo-user's comment here: http://tea.mathoverflow.net/discussion/947/ballsandbins-type-problem-question-closed