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Find n from a ratio of permutations or combinations

This type of question, asked 3 times: 31 Jan 2023, 6 Apr 2023, 31 Jan 2024. All 3 questions below, oldest first.

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  1. Question 1

    31 Jan 2023

    178
    If  2n+1Pn−1: 2n−1Pn=11:21\ ^{2n + 1}P_{n - 1}:\ ^{2n - 1}P_{n}= 11:21, then n2+n+15n^{2}+ n + 15 is equal to:

  2. Question 2

    6 Apr 2023

    63
    If  2nC3: nC3=10:1\ ^{2n}C_{3}:\ ^{n}C_{3}= 10:1, then the ratio (n2+3n):(n2−3n+4)\left( n^{2}+ 3n \right):\left( n^{2}- 3n + 4 \right) is

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  3. Question 3

    31 Jan 2024

    167
    If for some m,n; 6Cm+2( 6Cm+1)+ 6Cm+2> 8C3m,n;\ ^{6}C_{m}+ 2\left( \ ^{6}C_{m + 1} \right)+\ ^{6}C_{m + 2}>\ ^{8}C_{3} and  n−1P3: nP4=1:8\ ^{n- 1}P_{3}:\ ^{n}P_{4}= 1:8, then  nPm+1+ n+1Cm\ ^{n}P_{m+ 1}+\ ^{n+ 1}C_{m} is equal to

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