Algebra

Multinomial Theorem

The general term in the expansion of (𝑎+𝑏𝑥+𝑐𝑥²+⋯)^𝑛 is 𝑛(𝑛−1)(𝑛−2)⋯(𝑝+1)𝑞!𝑟!𝑠!⋯𝑎^𝑝𝑏^𝑞𝑐^𝑟𝑑^𝑠⋯𝑥^{𝑞+2𝑟+3𝑠+⋯}, where 𝑝+𝑞+𝑟+𝑠+⋯=⋯, and the number of terms 𝑝, 𝑞, 𝑟, ⋯ corresponds to the number of terms in the given multinomial.
𝑝 is integral, fractional, or negative, according as 𝑛 is one or the other.
If 𝑛 be an integer, may be written 𝑛𝑝!𝑞!𝑟!𝑠!𝑎^𝑝𝑏^𝑞𝑐^𝑟𝑑^𝑠⋯𝑥^{𝑞+2𝑟+3𝑠} Deduced from the Binomial Theorem.

Examples

To write the coefficient of 𝑎³𝑏𝑐⁵ in the expansion of (𝑎+𝑏+𝑐+𝑑)¹⁰. Here put 𝑛=10, 𝑥=1, 𝑝=3, 𝑞=1, 𝑟=5, 𝑠=0. Result: 10!3!5!=7⋅8⋅9⋅10

Examples

To obtain the coefficient of 𝑥⁸ in the expansion of (1−2𝑥+3𝑥²−4𝑥³)⁴. Here 𝑎=1, 𝑏=−2, 𝑐=3, 𝑑=−4, 𝑝+𝑞+𝑟+𝑠=4 𝑞+2𝑟+3𝑠=8 Possible values: 𝑝𝑞𝑟𝑠 1012 0202 0121 0040 The numbers 1, 0, 1, 2 are particular values of 𝑝, 𝑞, 𝑟, 𝑠 respectively, which satisfy the two equations given above. 0, 2, 0, 2 are another set of values which also satisfy those equations; and the four rows of numbers constitute all the solutions. In forming these rows always try the highest possible numbers on the right first. Now substitute each set of values of 𝑝, 𝑞, 𝑟, 𝑠 in the formula successively, 4!2!1¹(−2)⁰3¹(−4)²=576 4!2!2!1⁰(−2)²3⁰(−4)²=384 4!2!1⁰(−2)¹3²(−4)¹=864 4!4!1⁰(−2)⁰3⁴(−4)⁰=84 Result 1905

Examples

Required the coefficient of 𝑥⁴ in (1+2𝑥−4𝑥²−2𝑥³)⁻¹² Here 𝑎=1, 𝑏=2, 𝑐=−4, 𝑑=−2, 𝑛=−12; and the two equations are 𝑝+𝑞+𝑟+𝑠=−12 𝑞+2𝑟+3𝑠=4 Possible values: 𝑝𝑞𝑟𝑠 −52101 −52020 −72210 −92400 Employing the formula, the remainder of the work stands as follows: −12−321−5221(−4)0(−2)1=−3 12!−12−321−5220(−4)2(−2)0=6 12!−12−32−521−7222(−4)1(−2)0=15 14!−12−32−52−721−9224(−4)0(−2)0=358 Result: 2238

Examples

The number of terms in the expansion of the multinomial (𝑎+𝑏+𝑐+ to 𝑛 terms)^𝑟 is the same as the number of homogeneous products of 𝑛 things of 𝑟 dimensions.

Examples

The greatest coefficient in the expansion of (𝑎+𝑏+𝑐+ to 𝑚 terms)^𝑛, 𝑛 being an integer is 𝑛!(𝑞!)^𝑚(𝑞+1)^(𝑘), where 𝑞𝑚+𝑘=𝑛 Proof: By making the denominator in previous equation as small as possible. The notation is same as in Permutations, Combinations.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive