**•** How many poker hands are possible?

That is, how many possible sets of five cards can I draw from a 52-card deck?

52 possible choices for the first card, 51 for the second, …, 48 choices for the fifth. Altogether 52·51·50·49·48 ways of drawing the 5 cards.
But the order in which I pick up them doesn't matter. The possible orders are 5·4·3·2. So the number of possible hands is:

(52·51·50·49·48) / (5·4·3·2)
=
52·51·5·49·4
=
* *598* *960

**•** A biologist must do an experiment on 60 mice, chosen from the 100 available. In how many ways can he make the choice?

Cbin(100,60) = Cbin(100,40) = 100/40*99/39/*...61/1. We do the calculations with the computer:

choose(100,40) provides 1.374623·10^{28}, i.e. about 1.4·10^{28}. A huge number!

**•** y(0)** **=** **1, y(n+1)** **=** **(y(n)** **+** **A/y(n))** **/** **2, y(n) → **?**

A = 16; y = 1

y = (y+A/y)/2; y

# 8.5

y = (y+A/y)/2; y

# 5.191176

y = (y+A/y)/2; y

# 4.136665

# ...

y = (y+A/y)/2; y

# 4** √A**

**•** Check with the computer that for positive integer** n **
1³ + 2³ + 3³ +…+

Let's check with

f = function(n) (n*(n+1)/2)^2; g = function(n) sum((1:n)^3) for(i in 1:30) print(c(i, f(i), g(i))) # 1 1 1 # 2 9 9 # 3 36 36 # 4 100 100 # 5 225 225 # ... # 29 189225 189225 # 30 216225 216225