Question 179
There are $m$ boys and $n$ girls ($m, n \ge 2$) took part in a festival of singing couples. Each day, there are couples sang songs in a presentation. Each couple had a boy and a girl. In a presentation, every one sang at least one song, and a boy sang a duet with a girl not more than one time. Two presentations were different if there was at least one couple sang in exactly one of these two presentations. The festival ended when every possible presentation taken and each presentation taken exactly one time.
a) A presentation was dependent on $X$ if in the case $X$ was excluded together with the songs he/she sang then there was also another child that did not sing any song. Prove that, the number of presentations depending on $X$ and has odd number of songs is equal to the number of presentations depending on $X$ and has even number of songs.
b) Prove that we can arrange all presentations in a sequence of days so that two consecutive presentations' song numbers have different parities.
a) A presentation was dependent on $X$ if in the case $X$ was excluded together with the songs he/she sang then there was also another child that did not sing any song. Prove that, the number of presentations depending on $X$ and has odd number of songs is equal to the number of presentations depending on $X$ and has even number of songs.
b) Prove that we can arrange all presentations in a sequence of days so that two consecutive presentations' song numbers have different parities.