Personalized License Plate: Part 1
MTH109 – Mathematical Explorations
Colorado State University – Global Campus
Assuming repetitions are allowed, how many custom tag numbers are possible?
C(n, r) = (r + n – 1)! / (r! × (n – 1)!)
n = 36 and r = 6
Find (r + n – 1)!
(r + n – 1)! = (6 + 36 – 1)! = 41!
41!= 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35×36×37×38×39×40×41
= 3.3452526613163803e+49
Find r!
6! = 1×2×3×4×5×6 = 720
Find (n – 1)!
(n – 1)! = (36 – 1)! = 35!
35! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35
= 1.0333147966386144e+40
C(n, r) = (r + n – 1)! / (r! × (n – 1)!)
C(36, 6) = 3.3452526613163803e+49 / (720 × 1.0333147966386144e+40)
C(36, 6) = 4,496,388 = number of custom tag numbers with repetitions
Assuming repetitions are not allowed, how many custom tag numbers are possible?
C(n, r) = n! / (r! × (n – r)!)
n = 36 and r = 6
Find n!
36! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30×31×32×33×34×35×36
= 3.719933267899012e+41
Find r!
6! = 1×2×3×4×5×6 = 720
Find (n – r)!
(n – r)! = (36 – 6)! = 30!
30! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26×27×28×29×30
= 2.6525285981219103e+32
C(n, r) = n! / (r! × (n – r)!)
C(36, 6) = 3.719933267899012e+41 / (720 × 2.6525285981219103e+32)
C(36, 6) = 1,947,792 = number of custom tags with no repetitions
Assuming repetitions are not allowed and there must be exactly three letters, how many custom tag numbers are possible?
n = 6 and r = 3
C(n, r) = n! / (r! × (n – r)!)
Find n!
6! = 1×2×3×4×5×6 = 720
Find r!
3! = 1×2×3 = 6
Find (n – r)!
(n – r)! = (6 – 3)! = 3!
3! = 1×2×3 = 6
C(n, r) = n! / (r! × (n – r)!)
C(6, 3) = 720 / (6 × 6)
C(6, 3) = 20 = number of combinations where 3 of the positions will be letters
Letters that can be in any position, so long as there are 3 =
263 = 17,576
Numbers that can be in any position, so long as there are 3 =
103 = 1,000
20×17,576×1,000 = 351,520,000 = custom tags with exactly 3 letters
Assuming repetitions are not allowed and there must be exactly three letters, located at the beginning of the license plate, how many custom tag numbers are possible?
P(Ln,r)×P(Nn,r)
P(26,3)×P(10,3)
Letters
P(n, r) = n! / (n – r)!
Find n!
26! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23×24×25×26
= 4.0329146112660565e+26
Find (n – r)!
(n – r)! = (26 – 3)! = 23!
23! = 1×2×3×4×5×6×7×8×9×10×11×12×13×14×15×16×17×18×19×20×21×22×23 = 2.585201673888498e+22
P(n, r) = n! / (n – r)!
P(26, 3) = 4.0329146112660565e+26/2.585201673888498e+22
P(26, 3) = 15600
Numbers
P(n, r) = n! / (n – r)!
Find n!
10! = 1×2×3×4×5×6×7×8×9×10 = 3628800
Find (n – r)!
(n – r)! = (10 – 3)! = 7!
7! = 1×2×3×4×5×6×7 = 5040
P(n, r) = n! / (n – r)!
P(10, 3) = 3628800/5040
P(10, 3) = 720
P(26,3)×P(10,3) = 15600×720 = 11,232,000 = custom tags with 3 letters at the beginning
Assuming repetitions are not allowed, there must be exactly three letters, located at the beginning of the license plate, and that the letters can only be vowels, how many passwords are possible?
P(Vn,r)×P(Nn,r)
P(5,3)×P(10,3)
Vowels
P(n, r) = n! / (n – r)!
Find n!
5! = 1×2×3×4×5 = 120
Find (n – r)!
(n – r)! = (5 – 3)! = 2!
2! = 1×2 = 2
P(n, r) = n! / (n – r)!
P(5, 3) = 120/2
P(5, 3) = 60
Numbers
P(n, r) = n! / (n – r)!
Find n!
10! = 1×2×3×4×5×6×7×8×9×10 = 3628800
Find (n – r)!
(n – r)! = (10 – 3)! = 7!
7! = 1×2×3×4×5×6×7 = 5040
P(n, r) = n! / (n – r)!
P(10, 3) = 3628800/5040
P(10, 3) = 720
P(5,3)×P(10,3) = 60 × 720 = 43,200 = custom tags with exactly 3 vowels