Newton Discovered - Binomial Theorem Quiz

Newton discovered the binomial theorem in the year 1665.

The coefficients that appear in the binomial expansion are called binomial coefficients. These coefficients represent the numerical values that multiply each term in the expansion. They are calculated using combinations and represent the number of ways to choose a certain number of elements from a set. The binomial coefficients are denoted by the symbol "n choose k", where n is the total number of elements and k is the number of elements being chosen.

Free marks.

The given expression (r+1)th term = nCr an-r br is not necessarily false. It is a general formula to calculate the (r+1)th term in a binomial expansion, where nCr represents the binomial coefficient and an-r br represents the terms in the expansion. However, without any specific values or context provided for n, r, a, and b, it is not possible to determine the truth or falsity of the statement. Therefore, the answer is "Not False" as it cannot be definitively proven false without more information.

Newton was born in 1943, Pascal was born in 1923.

The statement "The number of permutations is not r! times as much as the number of combinations in nCr" is not true. In fact, the number of permutations is exactly r! times the number of combinations in nCr. Permutations refer to the arrangement of objects in a specific order, while combinations refer to the selection of objects without considering the order. Therefore, the number of permutations will always be greater than the number of combinations, and the ratio between them is r!.

The fifth term of the expansion of (x+2)^5 can be found using the binomial theorem. The general term of the expansion is given by C(5, k) * x^(5-k) * 2^k, where C(5, k) represents the binomial coefficient. To find the fifth term, we need to find the value of k that satisfies the condition (5-k) = 4. Solving this equation gives k = 1. Plugging this value into the general term formula, we get C(5, 1) * x^(5-1) * 2^1 = 5 * x^4 * 2 = 10x^4. Therefore, the fifth term of the expansion is 10x^4, which is equivalent to 16x.

The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results. Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.
http://upload.wikimedia.org/wikipedia/commons/e/ea/Yanghui_triangle.gif

To find the coefficient of the term in x^5, we need to expand the given expression using the binomial theorem. The general term in the expansion is given by (5 choose k) * (x^2)^(5-k) * (-1/2x^3)^k. For the coefficient of x^5, we need to find the value of k that satisfies (5 choose k) * (x^2)^(5-k) * (-1/2x^3)^k = x^5. Simplifying this equation, we get (-1/2)^k * x^(15-5k) = x^5. Equating the exponents, we get 15-5k = 5, which gives k = 2. Plugging this value of k into the equation, we get (-1/2)^2 * x^5 = (-1/4) * x^5 = -1/4 * x^5. Therefore, the coefficient of x^5 is -1/4, which is equal to -0.25 or -2.5.

The earliest extant Chinese illustration of 'Pascal's Triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (详解九章算法)[1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian [2] who expounded it around 1100 AD, about 500 years before Pascal.

Only his contributions to MATHEMATICS are included.

The term independent of x in the expansion can be found by looking for the term that does not contain any x. In this case, the term 9.072 does not have any x in it, so it is the term independent of x. The term 1134/125 does have an x in it, so it is not the term independent of x.