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Binomial coefficients

When solving combinatorial problems or calculating probabilities, terms called binomial coefficients are used. They are the coefficients that occur when developing the nth power of a binomial (a + b). They can be obtained from the so-called Pascal's number triangle - do my homework . The disadvantage is that this procedure is recursive, i.e. to determine the coefficients of (a+b)n, those of (a+b)n-1 must be known.

Therefore, an explicit definition of the binomial coefficients is given here, some calculation rules are made plausible, and the binomial theorem is formulated in general terms.

In the computational solution of combinatorial problems or in the calculation of probabilities, terms called binomial coefficients are used - geometry problem solver . They are the coefficients that occur when developing the nth power of a binomial (a + b). They can be obtained from the so-called Pascal's number triangle. The disadvantage, however, is that the procedure is recursive, i.e. to determine the coefficients of (a+b)n, those of (a+b)n-1 must be known. The following definition is therefore more advantageous:

The binomial coefficient (n k) (pronounced: n over k) is understood to be the

following expression:

(n k)=(n⋅(n-1)⋅(n-2)⋅...⋅[n-(k-1)])/(1⋅2⋅3⋅...⋅k) with n, k∈N and n≥k. It is (n 0)=1.

Note: The number of factors in the numerator and denominator of a binomial coefficient is the same.

Examples:

(10 3)=(10⋅9⋅8)/(1⋅2⋅3)=120

(90 2)=(90⋅89)/(1⋅2)=4 005

Using the factorial notation, the definition of the binomial coefficients can also be stated as follows:

(n k)=(n !)/(k !⋅(n-k) !) with n, k∈N and n≥k.

It can be seen immediately that (n1)=n holds.

Two further rules for calculating with binomial coefficients can be made plausible with the help of the Pascal number triangle - geometry homework help - and proven using the general definition.

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