Let ten balls be given. How many ways are there to put these into four boxes such that three boxes have capacity three and one has capacity four? How many possibilities are there if we suppose that from the ten balls seven is blue, three is red, and the red balls cannot share a box?

Show that the $r$-Eulerian numbers have the following special values
$$
\begin{aligned}
\left\langle\begin{array}{c}
n \
0
\end{array}\right\rangle_r & =r ! \
\left\langle\begin{array}{l}
n \
n
\end{array}\right\rangle_r & =r ! r^n \
\left\langle\begin{array}{c}
n \
n-1
\end{array}\right\rangle_r & =r !\left[(r+1)\left((r+1)^n-r^n\right)-n r^n\right] .
\end{aligned}
$$

Prove that the generating function of the hyperharmonic numbers is
$$
\sum_{n=0}^{\infty} H_n^r x^n=-\frac{\ln (1-x)}{(1-x)^r} .
$$
(The exponential generating function is more complicated, see [181, 185] for the details with respect to the ordinary harmonic numbers and $[425,421]$ for the exponential generating function of the hyperharmonic numbers.)

Based on Section 7.2 , prove the inequality
$$
\frac{(k+r)^n}{k !}-\frac{(k-1+r)^n}{(k-1) !}<\left{\begin{array}{l} n+r \ k+r \end{array}\right}_r<\frac{(k+r)^n}{k !} $$ for all $n \geq m>0$.

The game of Kayles. Kayles is an old English name for skittles or (bowling) pins. Two players are confronted with a row of pins. Their skill is such that they can knock down any one pin or any two adjacent ones. As usual the player who knocks down the last pin is the winner. Denoting pins by $\$$, a game might go as follows: $\$ \$ \$ \$ \$ \rightarrow \$ \$ _\$ \$ \rightarrow \$ _\$ \$ \$ \rightarrow _\$ \$ \$ \rightarrow _\$ _\$$ $\rightarrow \quad \$$ so that the second player wins this game. Who should win the game which starts $\$ \$ \$ \$ \$ \$ \$ ?$

Cram. This game is played by two persons on a $4 \times 5$ grid of squares. Players alternate putting a domino on the board so that it covers exactly two squares and does not overlap any already on the board. As usual, the last player to move is the winner.Who wins with best play and how?