(i) Prove that : \( { }^{n} P_{r}={ }^{n-1} P_{r}+r \cdot{ }^{n-1} P_{r-1} \) (ii) If \( { }^{20...
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0(i) Prove that : \( { }^{n} P_{r}={ }^{n-1} P_{r}+r \cdot{ }^{n-1} P_{r-1} \)
(ii) If \( { }^{20} \mathrm{C}_{\mathrm{r}+2}={ }^{20} \mathrm{C}_{2 \mathrm{r}-3} \) find \( { }^{12} \mathrm{C}_{\mathrm{r}} \)
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(iii) Prove that \( { }^{n-1} C_{3}+{ }^{n-1} C_{4}{ }^{n} C_{3} \) if \( n7 \).
(iv) Find \( \mathrm{r} \) if \( { }^{15} \mathrm{C}_{3 \mathrm{r}}={ }^{15} \mathrm{C}_{\mathrm{r}+3} \)
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