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प्रश्न
Find m and n if `""^(("m" + "n"))"P"_2` = 56 and `""^(("m" - "n"))"P"_2` = 12
उत्तर
`""^(("m" + "n"))"P"_2` = 56
∴ `(("m" + "n")!)/(("m" + "n" - 2)!) = 56`
∴ `(("m" + "n")("m" + "n" - 1)("m" + "n" - 2)!)/(("m" + "n" - 2)!)` = 56
∴ (m + n) (m + n – 1) = 8 × 7
Comparing on both sides, we get
(m + n) = 8 ......(i)
Also `""^(("m" - "n"))"P"_2` = 12
∴ `(("m" - "n")!)/(("m" - "n" - 2)!)` = 12
∴ `(("m" - "n")("m" - "n" - 1)("m" - "n" - 2)!)/(("m" - "n" - 2)!)` = 12
∴ (m – n) (m – n – 1) = 4 × 3
Comparing on both sides, we get
∴ m – n = 4 ......(ii)
Adding (i) and (ii), we get
2m = 12
∴ m = 6
Substituting m = 6 in (ii), we get
6 − n = 4
∴ n = 2
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