Help I don't know what to do here

You are given the 4 x 4 grid below.

(a) Find the number of ways of placing 8 counters in the squares (at most one counter per square), so that each row contains exactly two counters.

(b) Find the number of ways of placing 12 counters in the squares (at most one counter per square), so that each column contains exactly three counters.

Guest Apr 18, 2023

#1**0 **

I am going to assume that counters are distinguishable. Select 2 squares from each row to hold the counters: C(4,2)^2=6^4=1296.

Then, to decide the order of the counters C(8,2)*C(6,2)*C(4,2)*C(2,2) *2*2*2*2 to decide the order of the counters in each row. That is equal to 40320.

Note that that is also 8!, which is a simpler way to do this. Then, multiply to get **52254720**.

For the second one, choose one square in each column not to use: 4*4*4*4=256.

Then multiply by 12! to get **122624409600**.

gb1falcon May 2, 2023