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a) A pizza restaurant offers a choice of n ≥ 0 different toppings. (i) How many different pizzas are possible using 5 different toppings? (ii) Show, using the pizza restaurant scenario, and combinatorial proof that \left(\begin{array}{l}

n \\

0

\end{array}\right)+\left(\begin{array}{l}

n \\

1

\end{array}\right)+\left(\begin{array}{l}

n \\

2

\end{array}\right)+\ldots+\left(\begin{array}{l}

n \\

n

\end{array}\right)=2^{n} Ruth is organising a dinner party for 6 people in a small flat. She has 15 different friends but can only invite 6 of them due to space and chairs (i) In how many ways can she choose 6 friends to invite? (ii) In how many ways can she choose 6 friends if she also wishes to take the seating arrangement into consideration? ) Eight people leave their luggage at a check-in desk at a hotel. In how many ways can their luggage be returned to them so that (I) no person receives his own luggage (II) at least one of the people receives their own luggage (III) at least two of the people receive their own luggage (IV) exactly four people receive their own luggage In a survey of 80 people that had mild Covid, questions were asked regarding raised temperature, a cough and a loss of taste and/or smell. It was found that all people had at least one of the three symptoms and 15 people had all three symptoms. In fact, 45 had a temperature, 50 had a cough and 37 had lost their-sense of taste and/or smell. Use the principle of inclusion and exclusion for counting to determine how many people had exactly 2 of these symptoms.

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