An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Use the binomial coefficient formula: ( k n ) = k ! ( n − k )! n ! .
Substitute n = 3 and k = 2 into the formula: ( 2 3 ) = 2 ! ( 3 − 2 )! 3 ! .
Calculate the factorials: 3 ! = 6 , 2 ! = 2 , and 1 ! = 1 .
Simplify the expression: 2 × 1 6 = 3 . The number of ways is 3 .
Explanation
Understand the problem and provided data We are asked to find the number of ways to get k = 2 sixes among n = 3 rolls. We can use the binomial coefficient formula to calculate this. The formula is given by: ( k n ) = k ! ( n − k )! n ! where n is the total number of trials (rolls) and k is the number of successes (sixes).
Apply the binomial coefficient formula We are given n = 3 and k = 2 . Plugging these values into the binomial coefficient formula, we get: ( 2 3 ) = 2 ! ( 3 − 2 )! 3 ! Now, we need to calculate the factorials.
Calculate the factorials and simplify Recall that n ! = n × ( n − 1 ) × ( n − 2 ) × ... × 1 . So we have: 3 ! = 3 × 2 × 1 = 6 2 ! = 2 × 1 = 2 ( 3 − 2 )! = 1 ! = 1 Substituting these values into the expression, we get: ( 2 3 ) = 2 × 1 6 = 2 6 = 3 Therefore, there are 3 ways to get 2 sixes among 3 rolls.
State the final answer Thus, the number of ways to get k = 2 sixes among n = 3 rolls is 3.
Examples
Suppose you are flipping a coin 3 times and want to know how many ways you can get exactly 2 heads. This is a binomial probability problem where n=3 (number of flips) and k=2 (number of heads). The calculation ( 2 3 ) = 3 tells you there are 3 possible sequences: HHT, HTH, and THH.
