Probability: How many ways can a person select 4 programs from a television guide of 16 programs?
To find the number of ways a person can select 4 programs from a television guide of 16 programs, we can use the concept of combinations. The formula to calculate combinations is given as:
C(n, r) = n! / (r!(n - r)!)
Where:
C(n, r) represents the number of combinations of n items taken r at a time,
n! represents the factorial of n (n factorial),
r! represents the factorial of r, and
(n - r)! represents the factorial of (n - r).
In this case, we want to find the number of ways to select 4 programs from a total of 16 programs. Applying the formula, we have:
C(16, 4) = 16! / (4!(16 - 4)!)
Calculating the factorial values:
16! = 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 20,922,789,888,000
4! = 4 × 3 × 2 × 1 = 24
(16 - 4)! = 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479,001,600
Substituting the factorial values back into the formula:
C(16, 4) = 20,922,789,888,000 / (24 × 479,001,600)
Dividing the numerator by the denominator:
C(16, 4) ≈ 20,922,789,888,000 / 11,502,912,000
Simplifying the division:
C(16, 4) ≈ 1,820
Therefore, there are approximately 1,820 ways a person can select 4 programs from a television guide containing 16 programs.
To determine the number of ways a person can select 4 programs from a television guide of 16 programs, you can use the concept of combinations. The number of combinations can be calculated using the formula:
nCr = n! / r!(n-r)!
Where n is the total number of programs (16 in this case) and r is the number of programs to be selected (4 in this case). The exclamation mark (!) denotes factorial.
Plugging in the values, we have:
16C4 = 16! / 4!(16-4)!
= 16! / 4!12!
Next, let's calculate the factorials:
16! = 16 x 15 x 14 x 13 x 12!
4! = 4 x 3 x 2 x 1
Simplifying the expression:
16! / 4!12! = (16 x 15 x 14 x 13 x 12!) / (4 x 3 x 2 x 1 x 12!)
Canceling out the common terms (12!), we get:
16C4 = (16 x 15 x 14 x 13) / (4 x 3 x 2 x 1)
= 43680 / 24
= 1820
Therefore, there are 1820 different ways a person can select 4 programs from a television guide of 16 programs.