[tex]\huge\color{hotpink}\sf{Answer:}[/tex]
6. Seven marbles are chosen from 10 marbles of different colors. They are lined up into two lines in which the first line has 2 marbles and the second line has 5 marbles.
a. In how many ways can this be done?
[tex]\begin{gathered}_{7}P_{2} \cdot _{5}P_{5}\\42 \cdot 120\\=5040\end{gathered} [/tex]
b. If the red and the blue marbles must be in the first line, how many possible arrangements are there?
[tex]\begin{gathered}2 \cdot _{5}P_{5}\\2 \cdot 120\\=240\end{gathered} [/tex]
7. There are 4 different jewelry cases. Each can hold only one piece of jewelry.
a. Marco wants to buy 4 different pieces of jewelry and put them in the cases. In how many ways can he do this?
[tex]\begin{gathered}_{4}P_{4}\\=24\end{gathered}[/tex]
b. Marco now has only 2 different pieces of jewelry. Find the number of possible ways he can put them in the cases.
[tex]\begin{gathered}\frac{4!}{2!} \text{or} _{4}P_{2}\\= 12\end{gathered} [/tex]
c. Marco finally obtains the 4 pieces. However, he selects two of them to place in jewelry cases. What is the number of possible ways he can do this?
[tex]\begin{gathered}_{4}P_{2}\\=12\end{gathered} [/tex]
8. In a social dance course, there are 6 boys and 6 girls. Jack and Joe are two of them.
a. If a boy paired up with a girl for dancing, how many possible arrangements are there?
[tex]\begin{gathered}6 \cdot 6\\= 36\end{gathered} [/tex]
b. The students are required to demonstrate the dance, one pair at a time.
[tex]\begin{gathered}_{6}P_{6}\\=720\end{gathered}[/tex]
Suppose Jack and his partner do not want to be the first pair.
c. Find the number of orders of the demonstration sequence.
[tex]\begin{gathered}6! - 5!\\720 - 120\\= 600\end{gathered} [/tex]
d. In addition, if Joe and his partner do not want to be the last pair, find the number of orders.
[tex]\begin{gathered}6! - 5! - 5!\\720 - 120 - 120\\= 480\end{gathered} [/tex]
[tex]\huge\color{hotpink}\sf{Hikari\:Squad}[/tex]
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Alisa10
