<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>
Answer: B
Explanation:
We can substitute burgers for m and drinks for p, since all burgers are the same price and drinks are the same price, substitute the 3 burgers and 2 drinks into
3m+2p
The only right answer for this equation is $13.50, substitute the 5m+4p=$23.50 and your only right answer is B
Find completion to question in comment section.
Answer:
D. One of the jellybeans that slipped out was orange and one was black
Step-by-step explanation:
We calculate the option with the highest probability of occurrence :
Total number of jellybean = 75
n(T) =75
n(Pink) = 8
n(red) = 22
n(Orange) = 17
n(green) = 8
n(white) = 6
n(black) = 4
We assume that the jelly beans must have slipped out one after the other.
Evaluating the options :
A.)
P(pink) and P(white)
8/75 * 6/74 = 0.0086486
B.)
P(green) and P(green)
8/75 * 7/74 = 0.0100900
C.)
P(white) and P(white)
6/75 * 5/74 = 0.0054054
D.)
P(orange) and P(black)
17/75 * 4/74 = 0.0122522
From the probability values obtained, the highest is D. Hence, the most likely to have occurred is One of the jellybeans that slipped out was orange and one was black
When you divide 2872 by 76 the remainder is 7
