Exams Joint Entrance Examination Mathematics Permutations and Combinations Question 39 Question 39 of 278 Medium MCQ
If for some m , n ; 6 C m + 2 ( 6 C m + 1 ) + 6 C m + 2 > 8 C 3 m,n;{}^{6}{C}_{m}+2({}^{6}{C}_{m+1})+{}^{6}{C}_{m+2}>{}^{8}{C}_{3} m , n ; 6 C m + 2 ( 6 C m + 1 ) + 6 C m + 2 > 8 C 3 and n − 1 P 3 : n P 4 = 1 : 8 {}^{n-1}{P}_{3}:{}^{n}{P}_{4}=1:8 n − 1 P 3 : n P 4 = 1 : 8 , then n P m + 1 + n + 1 C m {}^{n}{P}_{m+1}+{}^{n+1}{C}_{m} n P m + 1 + n + 1 C m is equal to
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\[ 6 C m + 2 ( 6 C m + 1 ) + 6 C m + 2 > 8 C 3 7 C m + 1 + 7 C m + 2 > 8 C 3 8 C m + 2 > 8 C 3 ∴ m = 2 And n − 1 P 3 : n P 4 = 1 : 8 ( n − 1 ) ( n − 2 ) ( n − 3 ) n ( n − 1 ) ( n − 2 ) ( n − 3 ) = 1 8 ∴ n = 8 ∴ n P m + 1 + n + 1 C m = 8 P 3 + 9 C 2 = 8 × 7 × 6 + 9 × 8 2 = 372 \begin{aligned}
& { }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3 \\
& { }^7 \mathrm{C}_{\mathrm{m}+1}+{ }^7 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3 \\
& { }^8 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3 \\
& \therefore \mathrm{m}=2 \\
& \text { And }{ }^{\mathrm{n}-1} \mathrm{P}_3:{ }^n \mathrm{P}_4=1: 8 \\
& \frac{(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)}{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)}=\frac{1}{8} \\
& \therefore \mathrm{n}=8 \\
& \therefore{ }^{\mathrm{n}} \mathrm{P}_{\mathrm{m}+1}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{m}}={ }^8 \mathrm{P}_3+{ }^9 \mathrm{C}_2 \\
& =8 \times 7 \times 6+\frac{9 \times 8}{2} \\
& =372
\end{aligned} 6 C m + 2 ( 6 C m + 1 ) + 6 C m + 2 > 8 C 3 7 C m + 1 + 7 C m + 2 > 8 C 3 8 C m + 2 > 8 C 3 ∴ m = 2 And n − 1 P 3 : n P 4 = 1 : 8 n ( n − 1 ) ( n − 2 ) ( n − 3 ) ( n − 1 ) ( n − 2 ) ( n − 3 ) = 8 1 ∴ n = 8 ∴ n P m + 1 + n + 1 C m = 8 P 3 + 9 C 2 = 8 × 7 × 6 + 2 9 × 8 = 372 \] Answer the question to unlock