Andrew, a math teacher, needs to choose a team to send to a math compe

Question

Andrew, a math teacher, needs to choose a team to send to a math competition. \ Two of his best students, Maria and Nathan, refuse to work together. If Andrew has twelve students to choose from, and he must choose either Maria or Nathan so that he sends a strong team, how many possible five-person teams can he send?

A. 120
B. 252
C. 420
D. 792
E. 1320

A. 120
B. 252
C. 420
D. 792
E. 1320

quantumliner · Answer

Since either Maria or Nathan has to be part of the 5 member team,

Andrew can select 4 team members from the remaining 10 team members and add either Maria or Nathan as the 5th team members

To select 4 team members out of 10 = 10C4 = 10 ! / ( 6! * 4 !) = 210
Since either Maria or Nathan can be the 5th team member, total number of teams = 210 * 2 = 420

Answer is C. 420

ScottTargetTestPrep · Answer

We need to determine the number of ways Andrew can select a team with either Maria or Nathan.

If Maria is selected for the team but Nathan is not, the team can be selected in 10C4 ways (since Maria definitely makes the team and Nathan does not):

10C4 = (10 x 9 x 8 x 7)/4! = (10 x 9 x 8 x 7)/(4 x 3 x 2 x 1) = 10 x 3 x 7 = 210 ways

Since there are an additional 210 ways in which the team could be created (with Nathan and without Maria), Andrew can select the team in 210 + 210 = 420 ways.

Answer: C

Hero8888 · Answer

Hi,

Can you check please why I get wrong result? Here is how I calculate:
Favorable outcomes=Total outcomes-Unfavorable outcomes (when Maria and Nathan are together in the team)
Favorable outcomes=12!/7!5!-10!/3!7! (we have placed Maria and Nathan to the team of five and need to add other 3 people from the rest of 10 students) = 672

Thanks.

GMATYoda · Answer

Before you even get to the restriction on this problem, first identify that it is a combination problem - the order does not matter, but rather it's just about who's on the team. So prepare to use the Combinations formula: N!/K!(N−K!)

Whenever you encounter a combinations problem with restrictions, you have two major options:

That calculation will look like:

Maria is on the team, Nathan is not: 10 candidates for 4 remaining spots = 10!/4!6!=(10×9×8×7)/(4×3×2×1)

Nathan is on the team, Maria is not: the math is the same, with 10 candidates for 4 remaining spots. This is also 210

Therefore the correct answer is the sum of those two totals, 210+210=420

Choice C is correct.

castiel · Answer

Your Favorable outcomes also include the scenario when Nathan & Maria both are not present. So you need to subtract that as well from Total outcomes.
Favorable outcomes=12!/7!5!-10!/3!7! -10!/5!5!(when both Nathan & Maria aren't present)=420

GMATinsight · Answer

METHOD-1

Total Team of 5 out of 12 candidates = 12C5 = 792
Total Team of 5 out of 12 candidates  without  Maria and Nathan = 10C5 = 252
Total Team of 5 out of 12 candidates  with  Maria and Nathan both = 10C3 = 120

Favourable Outcomes = 792 - 252-120 = 420

Answer: Option C

METHOD-2

Total Team of 5 out of 12 candidates  with  Maria and  without  Nathan = 10C4 = 210
Total Team of 5 out of 12 candidates  with  Nathan and  without  Maria = 10C4 = 210

Favourable Outcomes = 210+210 = 420

Answer: Option C

arnkaprx · Answer

Why does simply 11C5 work here? (we can't choose from M or N, so we take one and create 5 team members from the 11 people left?)

Bunuel · Answer

We need to form teams of 5, which should include either Maria without Nathan, or Nathan without Maria.

As for your question, first ask yourself: who are you excluding from the 12 students to reduce the number to 11? If it's Maria, then you can form all 5-member teams from the remaining 11 students. However, this will not include teams where Maria is present and Nathan is not, but it will include teams where both Maria and Nathan are absent (for the team where Nathan is not chosen from 11). Similarly, if you exclude Nathan to reduce the count to 11, the issue remains the same. Therefore, it's necessary to use the approaches discussed earlier in this thread.

Hope it's clear.

luisdicampo · Answer

Deconstructing the Question

There are  12 students , including Maria and Nathan. We must choose a team of 5 students.

Maria and Nathan cannot be together, and at least one of them must be included. So we must choose exactly one of them.

Step-by-step

Case 1: Maria is selected and Nathan is excluded.

We choose the remaining 4 students from the other 10:

\binom{10}{4}

Case 2: Nathan is selected and Maria is excluded.

Again, choose the remaining 4 students from the other 10:

\binom{10}{4}

Total number of teams:

\binom{10}{4} + \binom{10}{4} = 2 \cdot \binom{10}{4}

= 2 \cdot 210 = 420

Answer C

Andrew, a math teacher, needs to choose a team to send to a math compe

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Andrew, a math teacher, needs to choose a team to send to a math compe

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