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Find the number of surjective (onto) functions $f$ from $\{1,2,3,\ldots,20\}$ to $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of 3 whenever $k$ is a multiple of 4.
Multiples of 4 in $\{1,\ldots,20\}$: $\{4,8,12,16,20\}$ โ 5 elements. Their images must be multiples of 3 in $\{1,\ldots,20\}$: $\{3,6,9,12,15,18\}$ โ 6 elements.
Assign the 5 multiples-of-4 to 5 of the 6 multiples-of-3 (injectively, since $f$ is onto): $6 \times 5 \times 4 \times 3 \times 2 = 6!/(6-5)! = 720$... but we need surjection on all 20 elements.
The remaining 15 domain elements map to the remaining 15 codomain elements (after fixing the 5 images above): $(15)!$ ways. But 1 multiple-of-3 is unused by the multiples-of-4, so it must be covered by remaining 15 elements.
Total $= 6! \times (15)! = (15)! \times 6!$
Matching logic:
- Lysine is an amino acid โ detected by ninhydrin (Q), which gives a purple colour with primary amines/amino acids.
- Furfural is an aldehyde from furan โ detected by 1-naphthol (P) via the Tollens-type or colour reaction specific to furfural.
- Benzyl alcohol (PhCH$_2$OH) is a primary alcohol โ detected by ceric ammonium nitrate (S), which gives a red colour with alcohols.
- Styrene (PhโCH=CH$_2$) has a C=C bond โ oxidized by KMnO$_4$ (R), which gets decolourized.
Correct match: (A)โ(Q), (B)โ(P), (C)โ(S), (D)โ(R)
The concept of 'Rule of Law' was popularized by which British jurist?
Albert Venn Dicey is famous for his exposition of the Rule of Law in his work on the UK Constitution.
Wavelength associated with freely falling body:
As it falls, velocity increases โ de Broglie wavelength ฮป = h/mv decreases.
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