If nCr denotes the binomial coefficient, then which of the following formula is correct?

Question

If nCr denotes the binomial coefficient, then which of the following formula is correct? (A)  n+1Cr-nCr-1=nCr  (B)  n+1Cr-n-1Cr=nCr  (C)  nCr+1-nCr-1=nCr  (D)  n-1Cr+nCr=n+1Cr

Tardigrade Admin · Accepted Answer

We know that, nCr+nCr-1=n+1Cr ⇒ nCr=n+1Cr-nCr-1

Q. If $^{n} C_{r}$ denotes the binomial coefficient, then which of the following formula is correct?

$^{n + 1} C_{r} -^{n} C_{r - 1} =^{n} C_{r}$

$^{n + 1} C_{r} -^{n - 1} C_{r} =^{n} C_{r}$

$^{n} C_{r + 1} -^{n} C_{r - 1} =^{n} C_{r}$

$^{n - 1} C_{r} +^{n} C_{r} =^{n + 1} C_{r}$

We know that, $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
$\Rightarrow$ $^{n} C_{r} =^{n + 1} C_{r} -^{n} C_{r - 1}$

Q. If nCr​ denotes the binomial coefficient, then which of the following formula is correct?

n+1Cr​−nCr−1​=nCr​

n+1Cr​−n−1Cr​=nCr​

nCr+1​−nCr−1​=nCr​

n−1Cr​+nCr​=n+1Cr​