Download App

Permutations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPermutations3 Marks
Question
prove that: $\frac{\text{n}!}{\text{(n-r)!r!}}+\frac{\text{n!}}{\text{(n-r+1)!}\text{(r-1)!}}= \frac{\text{(n+1)!}}{\text{r(n-r+1)!}}$
✓

Answer

We have, $\text{L.H.S.}=\frac{\text{n}!}{\text{(n-r)!r!}}+\frac{\text{n!}}{\text{(n-r+1)!}\text{(r-1)!}}$ $=\frac{\text{n!}}{(\text{n-r})!\text{r}\times\big[\text{(r-1)}!\big]}+\frac{\text{n!}}{\text{(n-r+1)}\big[(\text{n-r})!\big](\text{r-1})!}$ $=\frac{\text{n!}}{\text{(n - r)! }\times{\text{(r-1)!}}}\bigg[\frac{1}{\text{r}}+\frac{1}{\text{n-r+1}}\bigg]$ $=\frac{\text{n!}}{\text{(n - r)! }\times{\text{(r-1)!}}}\bigg[\frac{\text{n-r+1+r}}{\text{r(n-r+1)}}\bigg]$ $=\frac{\text{n!}}{\text{(n - r)! }\times{\text{(r-1)!}}}\bigg[\frac{\text{n+1}}{\text{r(n-r+1)}}\bigg]$ $=\frac{\text{(n+1)}\times\text{n!}}{\text{(n-r+1)!}\times\text{(n-r)}! \times\text{r}\times(\text{r-1})!}$ $=\frac{(\text{n+1}!)}{\text{(n-r+1)!}\times\text{r}!}$ $=\frac{\text{(n+1)!}}{\text{r!}\text{(n-r+1)}!}$ $\text{R.H.S}$ $\therefore$ L.H.S.= R.H.S.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from Permutations→Permutations — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The radius of a circle is 30cm. Find the length of an arc of this circle, if the length of the chord of the arc is 30cm.
→
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that $\frac{1}{{{p^2}}} = \frac{1}{{{a^2}}} + \frac{1}{{{b^2}}}$.
→
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
→
Find the magnitude, in radians and degrees, of the interior angle of a regular. Duodecagon.
→
If $\frac{1}{\text{a}+\text{b}},\frac{1}{2\text{b}},\frac{1}{\text{b}+\text{c}}$ are three consecutive terms ofan A.P., prove that a, b, c are the three consecutive terms of a G.P.
→
How many numbers of two digit are divisible by 3?
→
Find the equation of a circle, Which touches x-axis at a distance 5 from the origin and radius 6 units.
→
Find k such that $\text{k}+9,\text{k}-6$ and 4 from three consecutive terms of a G.P.
→
Show the following quadratic equation by factorization method: $3\text{x}-4\text{x}+\frac{20}{3}=0$
→
Differentiate the following functions with respect to x:$\frac{\sqrt{\text{a}}+\sqrt{\text{x}}}{\sqrt{\text{a}}-\sqrt{\text{x}}}$
→