• 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 = 2 598 960

• 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  1.374623·10²⁸, i.e. about 1.4·10²⁸. 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 +…+ n³ = (n·(n+1) / 2)²
Let's check with R. This can be demonstrated by induction.

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