An equal number of juniors and seniors are trying out for six spots

Hi,

We have an equal number of juniors and seniors. The problem states that we have 5 juniors, thus we only have 5 seniors. To make a team of 6 using a selection of 5 juniors and 5 seniors, the team must consist of at least 1 junior.

Does this make sense? To make a team of 6 seniors, you would need at least one more senior - the problem states that we only have 5.

I hope this helps -

If the team must have AT LEAST 4 seniors, then we must consider two possible cases:
Case i: The team has 4 seniors and 2 juniors
Case ii: The team has 5 seniors and 1 junior

Case i: The team has 4 seniors and 2 juniors
Since the order in which we select the seniors does not matter, we can use combinations.
STAGE 1: We can select 4 seniors from 5 seniors in 5C4 ways (= 5 ways)
STAGE 2: We can select 2 juniors from 5 juniors in 5C2 ways (= 10 ways)
By the Fundamental Counting Principle (FCP), we can complete both stages in (5)(10) ways = 50 ways

Aside: See the video below to learn how to quickly calculate combinations (like 5C2) in your head

Case ii: The team has 5 seniors and 1 junior
STAGE 1: We can select 5 seniors from 5 seniors in 5C5 ways (= 1 way)
STAGE 2: We can select 1 junior from 5 juniors in 5C1 ways (= 5 ways)
By the Fundamental Counting Principle (FCP), we can complete both stages in (1)(5) ways = 5 ways

TOTAL number of outcomes = 50 + 5 = 55

Answer: B

Cheers,
Brent

VIDEO ON CALCULATING COMBINATION IN YOUR HEAD

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