sensational
New member
- Joined
- Jun 4, 2015
- Messages
- 21
Q. Use the identity \(\displaystyle \, \displaystyle \binom{n}{r} \,=\, \dfrac{n!}{(n\, -\, r)!\, \times\, r!}\,\) to show that \(\displaystyle \, \displaystyle \binom{n\, +\, 1}{n\, -\, 2}\, =\, \dfrac{1}{6}\, (n^3\, -\, n).\)
I've viewed the answer but it makes me
.
A. \(\displaystyle \, \displaystyle \binom{n\, +\, 1}{n\, -\, 2}\, =\, \dfrac{(n\, +\, 1)!}{\bigg(\, (n\, +\, 1)\, -\, (n\, -\, 2)\, \bigg)!\, \times\, (n\, -\, 2)!}\)
. . . . .\(\displaystyle =\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)(n\, -\, 2)!}{3!\, \times\, (n\, -\, 2)!}=\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)\color{red}{{(n\, -\, 2)!}}}{3!\, \times\, \color{red}{{(n\, -\, 2)!}}}\)
. . . . . . . . . .\(\displaystyle =\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)}{6}\, =\, \dfrac{1}{6}\, (n^3\, -\, n)\, \) as required.
Could someone give me hints please by explaining the steps cause i don't know how this workout performed.
I've viewed the answer but it makes me
. A. \(\displaystyle \, \displaystyle \binom{n\, +\, 1}{n\, -\, 2}\, =\, \dfrac{(n\, +\, 1)!}{\bigg(\, (n\, +\, 1)\, -\, (n\, -\, 2)\, \bigg)!\, \times\, (n\, -\, 2)!}\)
. . . . .\(\displaystyle =\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)(n\, -\, 2)!}{3!\, \times\, (n\, -\, 2)!}=\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)\color{red}{{(n\, -\, 2)!}}}{3!\, \times\, \color{red}{{(n\, -\, 2)!}}}\)
. . . . . . . . . .\(\displaystyle =\, \dfrac{(n\, +\, 1)(n)(n\, -\, 1)}{6}\, =\, \dfrac{1}{6}\, (n^3\, -\, n)\, \) as required.
Could someone give me hints please by explaining the steps cause i don't know how this workout performed.
Last edited by a moderator: