If ^mP_r - ^{m-1}P_r = a cdot ^{m-1}P_s,then | vedclass

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(A) Given: $^mP_r - ^{m-1}P_r = a \cdot ^{m-1}P_s$ Using the formula $^nP_r = \frac{n!}{(n-r)!}$,we have: $\frac{m!}{(m-r)!} - \frac{(m-1)!}{(m-1-r)!}

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(A) Given: $^mP_r - ^{m-1}P_r = a \cdot ^{m-1}P_s$ Using the formula $^nP_r = \frac{n!}{(n-r)!}$,we have: $\frac{m!}{(m-r)!} - \frac{(m-1)!}{(m-1-r)!} = a \cdot \frac{(m-1)!}{(m-1-s)!}$ $\frac{m(m-1)!}{(m-r)(m-r-1)!} - \frac{(m-1)!}{(m-r-1)!} = a \cdot \frac{(m-1)!}{(m-s-1)!}$ $\frac{(m-1)!}{(m-r-1)!} \left( \frac{m}{m-r} - 1 \right) = a \cdot \frac{(m-1)!}{(m-s-1)!}$ $\frac{(m-1)!}{(m-r-1)!} \left( \frac{m - m + r}{m-r} \right) = a \cdot \frac{(m-1)!}{(m-s-1)!}$ $\frac{r \cdot (m-1)!}{(m-r)!} = a \cdot \frac{(m-1)!}{(m-s-1)!}$ Comparing the denominators,we get $m-r = m-s-1$,which implies $r = s+1$. Also,comparing the coefficients,we get $a = r$. Therefore,$a = s+1$,which gives $a - s = 1$.

If $^mP_r - ^{m-1}P_r = a \cdot ^{m-1}P_s$,then $a - s = \dots$

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