All combinations of three things selected from five | = | C^{5}_{3} | = | 5! -------------- 3! (5 - 3)! |
= | 10 |
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All permutations of three things selected from five | = | P^{5}_{3} | = | 5! -------------- (5 - 3)! |
= | 60 |
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Table 1. Expected number of bioassay tests required for isolation of two to five synergists in extracts of any number of compounds (n) and with initial fractionation into any number of fractions (f)^{a} | ||||
Two synergists | ||||
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Additive-combination tests = 1 + p_{2}C_{1}^{f} + p_{1+1}[SUM i=1 to 2]C_{i}^{f} + p_{2}[SUM i=1 to 2]C_{i}^{n} + p_{1+1}2C_{1}^{n} | ||||
Subtractive-combination tests = 1 + f + p_{2}C_{1}^{n} + p_{1+1}2C_{1}^{n} | ||||
Three synergists | ||||
Additive-combination tests = 1 + p_{3}C_{1}^{f} + p_{2+1}[SUM i=1 to 2]C_{i}^{f} + p_{1+1+1}[SUM i=1 to 3]C_{i}^{f} + p_{3}[SUM i=1 to 3]C_{i}^{n}+ p_{2+1}{ [SUM i=1 to 2]C_{i}^{n} + C_{1}^{n}} + p_{1+1+1}3C_{1}^{n} | ||||
Subtractive-combination tests = 1 + f + p_{3}C_{1}^{n} + p_{2+1}2C_{1}^{n} + p_{1+1+1}3C_{1}^{n} | ||||
Four synergists | ||||
Additive-combination tests = 1 + p_{4}C_{1}^{f} + p_{3+1}[SUM i=1 to 2]C_{i}^{f} + p_{2+2}[SUM i=1 to 2]C_{i}^{f} + p_{2+1+1}[SUM i=1 to 3]C_{i}^{f} + p_{1+1+1+1}[SUM i=1 to 4]C_{i}^{f} + p_{4}[SUM i=1 to 4]C_{i}^{n} + p_{3+1}{ [SUM i=1 to 3]C_{i}^{n} + C_{1}^{n}} + p_{2+2}{ [SUM i=1 to 2]C_{i}^{n} + [SUM i=1 to 2]C_{i}^{n}} + p_{2+1+1}{ [SUM i=1 to 2]C_{i}^{n} + 2C_{1}^{n}} + p_{1+1+1+1}4C_{1}^{n} | ||||
Subtractive-combination tests = 1 + f + p_{4}C_{1}^{n} + p_{3+1}2C_{1}^{n} + p_{2+2}2C_{1}^{n} + p_{2+1+1}3C_{1}^{n} + p_{1+1+1+1}4C_{1}^{n} | ||||
Five synergists | ||||
Additive-combination tests = 1 + p_{5}C_{1}^{f} + p_{4+1}[SUM i=1 to 2]C_{i}^{f} + p_{3+2}[SUM i=1 to 2]C_{i}^{f} + p_{3+1+1}[SUM i=1 to 3]C_{i}^{f} + p_{2+2+1}[SUM i=1 to 3]C_{i}^{f} + p_{2+1+1+1}[SUM i=1 to 4]C_{i}^{f} + p_{1+1+1+1+1}[SUM i=1 to 5]C_{i}^{f} + p_{5}[SUM i=1 to 5]C_{i}^{n} + p_{4+1}{ [SUM i=1 to 4]C_{i}^{n} + C_{1}^{n}} + p_{3+2}{ [SUM i=1 to 3]C_{i}^{n} + [SUM i=1 to 2]C_{i}^{n}} + p_{3+1+1}{ [SUM i=1 to 3]C_{i}^{n} + 2C_{1}^{n}} + p_{2+2+1}{2[SUM i=1 to 2]C_{i}^{n} + C_{1}^{n}} + p_{2+1+1+1}{ [SUM i=1 to 2]C_{i}^{n} + 3C_{1}^{n}} + p_{1+1+1+1+1}5C_{1}^{n} | ||||
Subtractive-combination tests = 1 + f + p_{5}C_{1}^{n} + p_{4+1}2C_{1}^{n} + p_{3+2}2C_{1}^{n} + p_{3+1+1}3C_{1}^{n} + p_{2+2+1}3C_{1}^{n} + p_{2+1+1+1}4C_{1}^{n} + p_{1+1+1+1+1}5C_{1}^{n} |
JOHN A. BYERS Department of Animal Ecology, Lund University, SE-223 62 Lund, Sweden Present address: Department of Plant Protection, Swedish University of Agricultural Sciences, SE-230 53 Alnarp, Sweden |
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