What are the coefficients for the binomial expansion of $(a+b)^3$?
A. $1,4,6,4,1$
B. 1, 1
C. $1,2,1$
D. $1,3,3,1$
Answer (1)
Expand ( a + b ) 3 using the binomial theorem.
Calculate the binomial coefficients: ( 0 3 ) = 1 , ( 1 3 ) = 3 , ( 2 3 ) = 3 , ( 3 3 ) = 1 .
The expansion is 1 a 3 + 3 a 2 b + 3 a b 2 + 1 b 3 .
The coefficients are 1 , 3 , 3 , 1 .
Explanation
Understanding the Problem We are asked to find the coefficients in the binomial expansion of ( a + b ) 3 . This is a standard problem that can be solved using the binomial theorem or by direct multiplication.
Applying the Binomial Theorem The binomial theorem states that ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k , where ( k n ) = k ! ( n − k )! n ! . For ( a + b ) 3 , the expansion is ( 0 3 ) a 3 b 0 + ( 1 3 ) a 2 b 1 + ( 2 3 ) a 1 b 2 + ( 3 3 ) a 0 b 3 .
Calculating the Coefficients Let's calculate the binomial coefficients:
( 0 3 ) = 0 ! ( 3 − 0 )! 3 ! = 1 ⋅ 3 ! 3 ! = 1
( 1 3 ) = 1 ! ( 3 − 1 )! 3 ! = 1 ⋅ 2 ! 3 ! = 1 ⋅ 2 ⋅ 1 3 ⋅ 2 ⋅ 1 = 3
( 2 3 ) = 2 ! ( 3 − 2 )! 3 ! = 2 ! ⋅ 1 ! 3 ! = 2 ⋅ 1 ⋅ 1 3 ⋅ 2 ⋅ 1 = 3
( 3 3 ) = 3 ! ( 3 − 3 )! 3 ! = 3 ! ⋅ 0 ! 3 ! = 3 ⋅ 2 ⋅ 1 ⋅ 1 3 ⋅ 2 ⋅ 1 = 1
Therefore, the binomial coefficients are 1 , 3 , 3 , 1 .
The Result The expansion is 1 a 3 + 3 a 2 b + 3 a b 2 + 1 b 3 . The coefficients are 1 , 3 , 3 , 1 .
Final Answer The coefficients for the binomial expansion of ( a + b ) 3 are 1 , 3 , 3 , 1 .
Examples
Binomial expansion is used in probability calculations, such as determining the likelihood of different outcomes in a series of independent trials. For instance, if you flip a coin three times, the binomial expansion of ( H + T ) 3 can help you calculate the probabilities of getting different combinations of heads (H) and tails (T). The coefficients tell you how many ways each combination can occur: 1 way to get 3 heads, 3 ways to get 2 heads and 1 tail, 3 ways to get 1 head and 2 tails, and 1 way to get 3 tails. This concept is fundamental in understanding probability distributions and making predictions in various scenarios, from games of chance to scientific experiments.
