Show that if n and j are integers with 0 ≤j ≤n, then, ( n j )=( n n-j ) Conc

step 1 N choose J is equivalent to the function n factorial over j factorial times n minus j factorial.

Show that if n and j are integers with 0 ≤j ≤n, then, ( ...

Show that if $n$ and $j$ are integers with $0 \leq j \leq n$, then,
$$
\left(\begin{array}{l}
n \\
j
\end{array}\right)=\left(\begin{array}{c}
n \\
n-j
\end{array}\right)
$$
Conclude that the Pascal triangle is symmetric with respect to a vertical line drawn from the topmost entry.

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Key Concepts

Binomial Coefficient

A binomial coefficient, commonly written as C(n, j) or (n choose j), represents the number of ways to select j items from a set of n distinct items without regard to order. It is defined using factorials and plays a central role in combinatorics, probability, and the binomial theorem.

Symmetry Property

The symmetry property of binomial coefficients states that selecting j items out of n is equivalent to selecting the complementary set of n - j items. This property, expressed as C(n, j) = C(n, n - j), reflects an inherent combinatorial balance and is often proved either algebraically using factorials or combinatorially by recognizing the dual interpretation in selection.

Pascal's Triangle

Pascal's Triangle is a geometric arrangement of binomial coefficients in a triangular layout where each number is the sum of the two numbers directly above it. The symmetry property of binomial coefficients is mirrored in the triangle's structure, resulting in a mirror image along its vertical center, which visually reinforces the combinatorial identity of the binomial coefficients.

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