This can be computed by hand using the distributive property of multiplication over addition, but it can also be done ( perhaps more easily ) with the multinomial theorem, which gives us a simple formula for any coefficient we might want.
12.
The distributive property also provides information about addition; by expanding the product ( 1 + 1 ) ( " a " + " b " ) in both ways, one concludes that addition is forced to be commutative.
13.
The distributive property ( cost to resist the locally dominant choice ) alone is not a network effect, for lack of any positive feedback, however the addition of bistability per individual, such as by a switching cost, qualifies as a network effect, by distributing this instability to the collective as a whole.
14.
They apply their understanding of models for multiplication ( i . e ., equal-sized groups, arrays, area models, equal intervals on the number line ), place value, and properties of operations ( in particular, the distributive property ) as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multidigit whole numbers.
15.
The properties of addition and multiplication have geometric counterparts; for instance, the distributive property of multiplication over addition can be justified geometrically for positive real numbers by representing a ( b + c ) as a rectangle whose sides are a and b + c, and noticing that we can also represent the same rectangle as a juxtaposition of two rectangles, axb and axc, giving a * b + a * c.
