DISCUSS ■ DISCOVER: Why Is the Same as ? This exercise explains why the binomial coefficients that appear in the expression of are the same as , the number of ways of choosing objects from objects. First, note that expanding a binomial using only the Distributive Property gives
(a) Expand using only the Distribution Property.
(b) Write all the terms that represent . These are all the terms that contain two ’s and three ’s.
(c) Note that the two ’s appear in all possible positions. Conclude that the number of terms that represent is .
(d) In general, explain why in the Binomial Theorem is the same as .
The expansion of the expression using only the Distributive Property.
The expression is and expansion by using the Distributive property is,
Expand the expression by using the Distributive property.
Use the Distributive Property again.
Again use the Distributive Property
All the terms that represent together.
The number of terms representing is .
The reason that in the Binomial theorem is same as .
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