Prove that m - nCr - mCr - mCr - 1 - n nC1 - mCr - 2 - nC2 - - - nCrwhere n- r - m and m- n- r are -ve integers-

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Standard XII

Mathematics

Middle Terms

Prove that ...

Question

Prove that
$^{m + n} C_{r} =^{m} C_{r} +^{m} C_{r - 1} \cdot n^{n} C_{1} +^{m} C_{r - 2} \cdot^{n} C_{2} + . . . . . +^{n} C_{r}$
where $n < r < m$ and $m, n, r$ are $+ v e$ integers.

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Solution

Multiply $(1 + x)^{m} a n d (1 + x)^{n}$ and the expression on R.H.S. . is coefficient of $x^{r}$ in the product
Now coefficient of $x^{r} i n (1 + x)^{m + n} i s^{m + n} C_{r}$

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^{m} C_{r} +^{m} C_{r - 1} \cdot^{n} C_{1} +^{m} C_{r - 2} \cdot^{n} C_{2} + . . . . . . . . +^{n} C_{r} =

Q. Prove that two +ive integers n, r cannot be found such that

^{n} C_{r},^{n} C_{r + 1},^{n} C_{r + 2} a n d^{n} C_{r + 3}

are in A.P.

Let r and n be positive integers such that $1 \leq r \leq n .$ Then prove the following :

(i) $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$
(ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$
(iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$
(iv) $^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}$

Q. If

n

and

r

are positive integers such that

r < n

, then

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

Q. Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
(a)

\frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} = \frac{n - r + 1}{r}

(b) n ·_{^{n −}}_¹C_r_{− 1} = (n − r + 1) ⁿC_r_{− 1}
(c)

\frac{{}^{n}C_{r}}{{}^{n - 1}C_{r - 1}} = \frac{n}{r}

(iv) ⁿC_r + 2 · ⁿC_r_{− 1} + ⁿC_r_{− 2} = ^{n +}²C_r.

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Prove that m+nCr=mCr+mCr−1⋅nnC1+mCr−2⋅nC2+.....+nCrwhere n<r<m and m,n,r are +ve integers.

Multiply (1+x)mand(1+x)n and the expression on R.H.S. . is coefficient of xr in the product Now coefficient of xrin(1+x)m+nism+nCr

Prove that
$^{m + n} C_{r} =^{m} C_{r} +^{m} C_{r - 1} \cdot n^{n} C_{1} +^{m} C_{r - 2} \cdot^{n} C_{2} + . . . . . +^{n} C_{r}$
where $n < r < m$ and $m, n, r$ are $+ v e$ integers.

Multiply $(1 + x)^{m} a n d (1 + x)^{n}$ and the expression on R.H.S. . is coefficient of $x^{r}$ in the product
Now coefficient of $x^{r} i n (1 + x)^{m + n} i s^{m + n} C_{r}$