Understanding Binomial Identity Proof

This video tutorial provides an algebraic proof for the binomial identity n choose k equals n minus 1 choose k minus 1 plus n minus 1 choose k. It begins with a review of the factorial definition of n choose k and explains the expansion of factorials. The proof is set up by applying the factorial definition to n minus 1 choose k minus 1 and n minus 1 choose k. The tutorial then simplifies the fractions by multiplying and factoring, leading to a common denominator. Finally, the proof is completed by showing that the expression simplifies to n choose k, thus proving the identity.

What is the binomial identity that is being proved in this lesson?

• n choose k equals n choose k plus 1

• n choose k equals n minus 1 choose k

• n choose k equals n minus 1 choose k minus 1 plus n minus 1 choose k

• n choose k equals n choose k minus 1

How is n choose k defined in terms of factorials?

• n factorial divided by n factorial

• n factorial divided by k factorial

• n factorial divided by n minus k factorial

• n factorial divided by the product of n minus k factorial and k factorial

What is the expanded form of k factorial?

• k times k factorial

• k times k minus one factorial

• k minus one times k factorial

• k factorial times n factorial

What is the first step in setting up the proof?

• Applying the factorial definition for n choose k

• Simplifying k factorial

• Applying the factorial definition for n minus one choose k minus one and n minus one choose k

• Expanding n factorial

What do you multiply the first fraction by during simplification?

• n minus k over n minus k

• k minus one over k minus one

• n over n

• k over k

What is the common denominator found during the simplification process?

• n minus k factorial

• The product of n minus k factorial and k factorial

• n factorial

• k factorial

What is factored out from the numerator in the final steps of the proof?

• n factorial

• k factorial

• n minus k factorial

• n minus one factorial

What does k plus n minus k simplify to in the numerator?

• k

• n

• n minus k

• k minus n

What is the final expression obtained in the proof?

• n factorial divided by n factorial

• n factorial divided by k factorial

• n factorial divided by the product of n minus k factorial and k factorial

• n factorial divided by n minus k factorial

What is the conclusion of the proof?

• n choose k equals n factorial

• n choose k equals n minus k factorial

• n choose k equals n choose k minus 1

• n choose k equals n minus 1 choose k minus 1 plus n minus 1 choose k