[Solved] $\text { Prove for all integers } n , k \text {, and } r \text { with } n \geq k \geq r \text { that } \left( \begin{array} { l } n \\ k \end{array} \right) \left( \begin{array} { l } k \\ r \end{array} \right) = \left( \begin{array} { l } n \\ r \end{array} \right) \left( \begin{array} { l } n - r \\ k - r \end{array} \right) \text {. }$ Dorothea Dix: A 19th-century advocate for the mentally ill who significantly improved the conditions of psychiatric hospitals in the United States. Treatment Of The Mentally Ill: The range of services, interventions, and supports provided to individuals with mental health conditions, aiming to improve their quality of life and functioning. Franz Anton Mesmer: A German physician who developed Mesmerism, considered a precursor to hypnosis, based on his theory of "animal magnetism." Magnetic Fluid: A hypothetical liquid that responds to magnetic fields, used often in science fiction or theoretical discussions for medical or technological applications.

Solved

question 4

Essay

$\text { Prove for all integers } n , k \text {, and } r \text { with } n \geq k \geq r \text { that } \left( \begin{array} { l } n \\k\end{array} \right) \left( \begin{array} { l } k \\r\end{array} \right) = \left( \begin{array} { l } n \\r\end{array} \right) \left( \begin{array} { l } n - r \\k - r\end{array} \right) \text {. }$

Definitions:

Dorothea Dix

A 19th-century advocate for the mentally ill who significantly improved the conditions of psychiatric hospitals in the United States.

Treatment Of The Mentally Ill

The range of services, interventions, and supports provided to individuals with mental health conditions, aiming to improve their quality of life and functioning.

Franz Anton Mesmer

A German physician who developed Mesmerism, considered a precursor to hypnosis, based on his theory of "animal magnetism."

Magnetic Fluid

A hypothetical liquid that responds to magnetic fields, used often in science fiction or theoretical discussions for medical or technological applications.