• 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
=
• 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 JavaScript (see). This can be demonstrated by induction.
s=0
for(n=1; n<=30; n=n+1) {
s=s+Math.pow(n,3);
document.write (n," - ", Math.pow(n*(n+1)/2,2), " ", s, "<br>") }
1 - 1 1
2 - 9 9
3 - 36 36
4 - 100 100
5 - 225 225
6 - 441 441
: . . .
29 - 189225 189225
30 - 216225 216225