Respuesta :
Answer:
We conclude that he coach can choose 252 lineups.
Step-by-step explanation:
We have three pure centers, four pure forwards, four pure guards and one swing- man who can play either guard or forward.
First case: swing- man play guard.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 1.
[tex]C_1^3\cdot C_2^4\cdot C_1^4=3\cdot \frac{4!}{2!(4-2)!}\cdot 4=12\cdot 6=72[/tex]
Second case: swing- man play forward.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 1, and from 4 from the guards to choose 2.
[tex]C_1^3\cdot C_1^4\cdot C_2^4=3\cdot 4\cdot \frac{4!}{2!(4-2)!}=12\cdot 6=72[/tex]
Third case: swing- man not play.
So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 2.
[tex]C_1^3\cdot C_2^4\cdot C_2^4=3\cdot \frac{4!}{2!(4-2)!}\cdot \frac{4!}{2!(4-2)!}=3\cdot 6 \cdot 6=108[/tex]
Now, we get 72+72+108=252.
We conclude that he coach can choose 252 lineups.
The number of ways the coach can start a lineup with one center, two forwards and two guards is 360.
How many possible lineups can the coach choose?
We will use combinations to find the possible lineups,
given to us
Number of pure centers = 3
Number of pure forwards = 4
Number of pure guards = 4
Number of swingmen = 1 (can play either guard or forward)
Case I: When the swingmen will play as guards,
Number of pure centers = 3
Number of pure forwards = 4
Number of pure guards = 5
Number of possibilities = Number of ways to choose pure center x Number of ways to choose pure forward x Number of ways to choose pure guard
[tex]\begin{aligned}\text{Number of possibilities} &= ^3C_1 \times ^4C_2 \times ^5C_2\\&=3 \times 6 \times 10\\& = 180\end{aligned}[/tex]
Case II: When the swingmen will play as forward,
Number of pure centers = 3
Number of pure forwards = 5
Number of pure guards = 4
Number of possibilities = Number of ways to choose pure center x Number of ways to choose pure forward x Number of ways to choose pure guard
[tex]\begin{aligned}\text{Number of possibilities} &= ^3C_1 \times ^5C_2 \times ^4C_2\\&=3 \times 10 \times 6\\& = 180\end{aligned}[/tex]
Total number of Possibilities
Total number of Possibilities = 180 + 180
= 360
Hence, the number of ways the coach can start a lineup with one center, two forwards and two guards is 360.
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