Question

1)

a) What is the minimum number of cards that you must pick out of a shuffled standard 52-card deck to guarantee that you will get at least one face card (King, Queen, or Jack)? Explain your answer. A standard 52 card deck contains 4 suits of 13 cards.

b) You are buying a box of 16 truffles as a birthday gift for a friend of yours. You will select which truffles to put in the box. The chocolate store has 20 different varieties of truffles who are all the same size and all cost the same. They have a large supply of all 20 varieties, so your choice will not be restricted by the store’s inventory. How many different combinations of truffles can you put in the box of 16? Explain your answer.

Answer 1

Question (1)

There is a shuffled standard 52-card deck

Number of face cards in each suit = 3 (King, Queen or Jack)

Number of suits = 4

So total Number of face cards = 12

Number of cards that are not face cards = 52-13 = 39

In the worst case scenario, we can pick the 39 cards that are not face cards in our first 39 picks

So the minimum number of cards that you must pick out of a shuffled standard 52-card deck to guarantee that you will get at least one face card = 40 (becuase i can pick 39 cards that are not face cards in the first 39 picks)

Question (2)

This can be done directly by applying the combinations formula, but here i will explain in detail about the procedure to arrive at the answer and next will also give you the formula too

The chocolate store has 20 different varieties of truffles who are all the same size and all cost the same

You are buying a box of 16 truffles as a birthday gift for a friend of yours

Number of different combinations of truffles can you put in the box of 16 is obtained as follows

The first truffle can be any of the 20 tuffles

The seocnd truffle can be any of the remaining19 tuffles

The third truffle can be any of the remaining 18 tuffles

The fourth truffle can be any of the remaining 17 tuffles

Likeiwse we repeat for all

The fiteenth truffle can be any of remaining 6 tuffles

The sixteenth truffle can be any of remaining 5 tuffles

So total number of ways this can be done is 20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5 ways

But the order is not important here becuase they may be in any order. So we need to divide the above obtained ways by 16! which is 16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1

So different combinations of truffles can you put in the box of 16 is

(20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5) / (16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)

= 20*19*18*17 / 4*3*2*1

= 116280 / 24

= 4845

So different combinations of truffles can you put in the box of 16 is 4845

The formula for selecting x items out of n items is which is n! / x! (n-x)!

So here we need to select 16 truffles from 20 varieties

So number of combinations of truffles can you put in the box of 16 is

=  20! / 16! (20-16)!

= 20! / 16! * 4!

= 20*19*18*17 / 4*3*2*1

= 116280 / 24

= 4845

So different combinations of truffles can you put in the box of 16 is 4845