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Jared will pick 3 friends to join him on a road trip. His friend group
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19 Jan 2023, 11:43
19 Jan 2023, 11:43
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Question Stats:
75% (02:06) correct
25% (01:57) wrong based on 16 sessions
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Jared will pick 3 friends to join him on a road trip. His friend group consists of 4 musicians and 3 poets. In how many different ways can Jared select his 3 traveling companions so that he has at least one musician and at least one poet among them?
(A) 16
(B) 18
(C) 30
(D) 36
(E) 84
(A) 16
(B) 18
(C) 30
(D) 36
(E) 84
Re: Jared will pick 3 friends to join him on a road trip. His friend group
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24 Jan 2023, 06:16
24 Jan 2023, 06:16
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Expert Reply
OE
Jared has to select either 2 musicians and 1 poet, or 1 musician and 2 poets. The number of ways he can select 2 musicians from among 4 is given by 4!/2!*2!=6. while the number of ways to select 1 poet from among 3 is 3. Thus, there are 6 x 3 = 18 ways to select 2 musicians and 1 poet. Similarly, the number of ways to select 2 poets out of 3 is 3!/2!*1!=3 while the number of ways to select 1 musician from among 4 is 4. This gives Jared another 4 x 3 = 12 ways to select his traveling companions. The total number of options Jared has is 18 + 12, which is 30.
Jared has to select either 2 musicians and 1 poet, or 1 musician and 2 poets. The number of ways he can select 2 musicians from among 4 is given by 4!/2!*2!=6. while the number of ways to select 1 poet from among 3 is 3. Thus, there are 6 x 3 = 18 ways to select 2 musicians and 1 poet. Similarly, the number of ways to select 2 poets out of 3 is 3!/2!*1!=3 while the number of ways to select 1 musician from among 4 is 4. This gives Jared another 4 x 3 = 12 ways to select his traveling companions. The total number of options Jared has is 18 + 12, which is 30.
Re: Jared will pick 3 friends to join him on a road trip. His friend group
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29 Dec 2023, 16:54
29 Dec 2023, 16:54
1
Another way to do it that I think requires less calculations:
# of total combinations if poet/musician didn't matter - # of combos with no poets - # of combos with no musicians = # of combinations with at least one musician and at least one poet.
# of total combinations if poet/musician didn't matter = 7 choose 3 = 35
# of combos with no poets = 3 choose 3 = 1
# of combos with no musicians = 4 choose 3 = 4
35-1-4 = 30
Video for more info/explanations: Search Tested Tutor Advanced Combinations
# of total combinations if poet/musician didn't matter - # of combos with no poets - # of combos with no musicians = # of combinations with at least one musician and at least one poet.
# of total combinations if poet/musician didn't matter = 7 choose 3 = 35
# of combos with no poets = 3 choose 3 = 1
# of combos with no musicians = 4 choose 3 = 4
35-1-4 = 30
Video for more info/explanations: Search Tested Tutor Advanced Combinations
