Question

Untitled-Notepad File Edit Format View Help Counting 1. Our student clubs are being particularly active. The aquatics club in particular is in training for an upcoming competition. For all parts of this question, working is required, including combinatoric/factorial notation as needed as well as 1nal answers. a) During training, there are several different types of laps that can be swuM: . freestyle arms only freestyle one arm only, left then right . torpedo/kickboard (straight kicking legs only) - backstroke breaststroke Assume that each type is undertaken at most once in a training session, over contiguous laps (i.e., so once you have completed your freestyle laps, you do not repeat freestyle). The number of laps for each lap type is unimportant, that is, 4 laps of freestyle followed by 2 laps of backstroke is the same as 3 laps of freestyle followed by 4 laps of backstroke i. How many different lap type patterns can be formed from those six types? її. How many patterns contain only three types (e.g., freestyle - arms only-one arm only)? iii. How many patterns with at least four different types start with torpedo/kickboard laps?

Answer 1

i)

Now, each pattern can be undertaken at most once.

We can include or exclude the lap from our pattern.

First calculating for only one style included.

Selecting a style can be done in ⁶C₁ways and arranging them can be done in 1! way.

So ⁶C₁x1!= 6 ways.

Now calculating for two style included.

Selecting a style can be done in ⁶C₂ ways and arranging them can be done in 2! way.

So ⁶C₂x2!= 30 ways.

Similarly for 3, 4, 5 and 6 styles included can be done in ⁶C₃x3!, ⁶C₄x4!, ⁶C₅x5!, ⁶C₆x6! ways

Adding them we get = ⁶C₁x1!+⁶C₂x2!+⁶C₃x3!+⁶C₄x4!+⁶C₅x5!+⁶C₆x6!=6+30+120+360+720+720=1956 ways.

ii)

Pattern containing only 3 types

3 types can be chosen in ⁶C₃ ways.

They can be arranged in 3! ways.

SO, patterns = ⁶C₃x3!=120 patterns.

iii)

Given, at least 4 different types and start with torpedo/kickboard laps.

For 4 different types laps,

Starting with torpedo/kickboard, now we need to chose and arrange 3 other lap patterns.

This can be done in ⁵C₃x3! =60 ways.

For 5 different types laps,

Starting with torpedo/kickboard, now we need to chose and arrange 4 other lap patterns.

This can be done in ⁵C₄x4! =120 ways.

Similarly for 6 laps, it can be done in ⁵C₅x5! =120 ways.

Adding them we get, 120+120+60 = 300 ways.

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