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1 answer:
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Explanation:
Each row in Pascal's triangle is a listing of the values of nCk = n!/(k!(n-k)!) for some fixed n and k in the range 0 to n. nCk is <em>the number of combinations of n things taken k at a time</em>.
If you consider what happens when you multiply out the product (a +b)^n, you can see where the coefficients nCk come from. For example, consider the cube ...
(a +b)^3 = (a +b)(a +b)(a +b)
The highest-degree "a" term will be a^3, the result of multiplying together the first terms of each of the binomials.
The term a^b will have a coefficient that reflects the sum of all the ways you can get a^b by multiplying different combinations of the terms. Here they are ...
- (a +_)(a +_)(_ +b) = a·a·b = a^2b
- (a +_)(_ +b)(a +_) = a·b·a = a^2b
- (_ +b)(a +_)(a +_) = b·a·a = a^2b
Adding these three products together gives 3a^2b, the second term of the expansion.
For this cubic, the third term of the expansion is the sum of the ways you can get ab^2. It is essentially what is shown above, but with "a" and "b" swapped. Hence, there are 3 combinations, and the total is 3ab^2.
Of course, there is only one way to get b^3.
So the expansion of the cube (a+b)^3 is ...
(a +b)^3 = a^3 + 3a^2b +3ab^2 +b^3 . . . . . with coefficients 1, 3, 3, 1 matching the 4th row of Pascal's triangle.
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In short, the values in Pascal's triangle are the values of the number of combinations of n things taken k at a time. The coefficients of a binomial expansion are also the number of combinations of n things taken k at a time. Each term of the expansion of (a+b)^n is of the form (nCk)·a^(n-k)·b^k for k =0 to n.
Answer:
Angle,
Step-by-step explanation:
We have, an A-frame house is 45 feet high and 32 feet wide. It is required to find the measure of the angle that the roof makes with the floor.
The attached figure shows the A frame. Let is the angle that the roof makes with the floor. So,
So, the angle of 70.4 degrees is the angle that the roof makes with the floor.
