Question

If for some \(m, n ; \,^6 C _m + 2 (^6C_{m+ 1})+ ^6 C _{m+ 2}> ^8 C_3\) and \(^{n-1}P_ 3 : \,^n P_ 4 = 1: 8 ,\) then \(^nP_{m+ 1}+^{n+ 1}C _m\) is equal to

(1) 380

(2) 376

(3) 384

(4) 372

Answer 1

 Correct option is (4) 372

\({ }^{6} \mathrm{C}_{\mathrm{m}}+2\left({ }^{6} \mathrm{C}_{\mathrm{m}+1}\right)+{ }^{6} \mathrm{C}_{\mathrm{m}+2}>{ }^{8} \mathrm{C}_{3}\)

\({ }^{7} \mathrm{C}_{\mathrm{m}+1}+{ }^{7} \mathrm{C}_{\mathrm{m}+2}>{ }^{8} \mathrm{C}_{3}\)

\({ }^{8} \mathrm{C}_{\mathrm{m}+2}>^{8} \mathrm{C}_{3}\)

\(\therefore \mathrm{m}=2\)

And \({ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8\)

\(\frac{(n-1)(n-2)(n-3)}{n(n-1)(n-2)(n-3)}=\frac{1}{8}\)

\(\therefore \mathrm{n}=8\)

\(\therefore{ }^{n} P_{m+1}+{ }^{n+1} C_{m}={ }^{8} P_{3}+{ }^{9} C_{2}\)

\(=8 \times 7 \times 6+\frac{9 \times 8}{2}\)

\(=372\)