smc-1082-30-Combinations in a Circle

Combinatorics, EXT1 A1 EQ-Bank 3

Four girls and four boys are to be seated around a circular table. In how many ways can the eight children be seated if:

there are no restrictions? (1 mark)

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the two tallest boys must not be seated next to each other? (1 mark)

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the two youngest children sit together? (1 mark)

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A group with 5 students and 3 teachers is to be arranged in a circle.

In how many ways can this be done if no more than 2 students can sit together?

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♦♦♦ Mean mark 13%.

Eight points , are arranged in order around a circle, as shown below.

How many triangles can be drawn using these points as vertices? (1 mark)

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How many pairs of triangles can be drawn, where the vertices of each triangle are distinct points? (2 marks)

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i.

COMMENT: In part (ii), divide by 2 to account for duplicate pairs.

ii.

In how many ways can the numbers 9, 8, 7, 6, 5, 4 be arranged around a circle? (1 mark)

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How many of these arrangements have at least two odd numbers together? (2 marks)

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ii.

Six men and six women are to be seated at a round table.

In how many different ways can they be seated if men and women alternate?

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♦ Mean mark 42%.

In how many ways can 6 people from a group of 15 people be chosen and then arranged
in a circle?

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A family of eight is seated randomly around a circular table.

What is the probability that the two youngest members of the family sit together?

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