<span>binomial </span>is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
<span>(a + b)4</span> <span>= a4 + 4a3b</span><span> + 6a2b2 + 4ab3 + b4</span>
<span>(a + b)5</span> <span>= a5 + 5a4b</span> <span>+ 10a3b2</span><span> + 10a2b3 + 5ab4 + b5</span>
Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.
Pascal's Triangle
We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
You can use this pattern to form the coefficients, rather than multiply everything out as we did above.
The Binomial Theorem
We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.
<span>Properties of the Binomial Expansion <span>(a + b)n</span></span><span><span>There are <span>\displaystyle{n}+{1}<span>n+1</span></span> terms.</span><span>The first term is <span>an</span> and the final term is <span>bn</span>.</span></span><span>Progressing from the first term to the last, the exponent of a decreases by <span>\displaystyle{1}1</span> from term to term while the exponent of b increases by <span>\displaystyle{1}1</span>. In addition, the sum of the exponents of a and b in each term is n.</span><span>If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.</span>
The probability that that a randomly selected student will buy a raffle ticket and win a prize = 0.03
Step-by-step explanation:
Step 1 :
Given,
The percentage of students buying the raffle ticket = 30%
The percentage that the student who bought the ticket wins the prize = 10%
We need to determined the probability that randomly selected student will buy a raffle ticket and win a prize.
Step 2 :
The probability that a student buys the raffle ticket = 
The probability that a student wins a prize = 
= 
The probability that a student who buys the ticket wins a price can be computed by taking the product of the above 2 probabilities.
= 
× = 
= 0.03
Step 3 :
Answer :
The probability that that a randomly selected student will buy a raffle ticket and win a prize = 0.03
Answer:
area formula: base x height 9x4=36
Step-by-step explanation:
Answer:
1 pink marble
Step-by-step explanation:
If Jamie selected 10 times the pink marble in a total of 420 tries, we can assume that the proportion of pink marble in the total number of marbles is:
10 / 420 = 1/42
So to predict a good number of pink marbles in the total 40 marbles, we just need to multiply the 40 marbles by the proportion found above:
pink marble = 40 * (1/42) = 0.9524
As this number needs to be a whole number, a good assumption is that there is 1 pink marble in the 40 marbles.
