7: The basket ball team has total 14 players : 3-1st year player,5-2nd year player and 6-3rd year player (a) in how many ways can the coach choose a starting line up(5 players) with at least one 1 st player. (b) in how many ways can he set up a starting lineup with two 2nd year and three 3rd year player?
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To solve both parts of this problem, we can use combinations. The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items to choose from, and r is the number of items to choose.
(a) To find the number of ways the coach can choose a starting lineup with at least one 1st-year player, we can subtract the number of lineups without any 1st-year players from the total number of lineups.
Total number of lineups = C(14, 5)
Number of lineups without any 1st-year player = C(11, 5)
Therefore, the number of lineups with at least one 1st-year player = C(14, 5) - C(11, 5)
(b) To find the number of ways the coach can set up a starting lineup with two 2nd-year players and three 3rd-year players, we need to choose 2 players from the 5 2nd-year players and 3 players from the 6 3rd-year players.
Number of ways to choose 2nd-year players = C(5, 2)
Number of ways to choose 3rd-year players = C(6, 3)
Therefore, the number of ways to set up the starting lineup = C(5, 2) * C(6, 3)
Now, let's calculate the answers.
(a) Number of lineups with at least one 1st-year player:
Total number of lineups = C(14, 5) = 2002
Number of lineups without any 1st-year player = C(11, 5) = 462
Number of lineups with at least one 1st-year player = 2002 - 462 = 1540
(b) Number of lineups with two 2nd-year players and three 3rd-year players:
Number of ways to choose 2nd-year players = C(5, 2) = 10
Number of ways to choose 3rd-year players = C(6, 3) = 20
Number of ways to set up the starting lineup = 10 * 20 = 200
Therefore,
(a) The coach can choose a starting lineup with at least one 1st-year player in 1540 different ways.
(b) The coach can set up a starting lineup with two 2nd-year players and three 3rd-year players in 200 different ways.