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EXERCISE 4.2.9. Let P be the set of people in a group, with |P| = p. Let C be a set of clubs formed by the people in this group, with |C| = c. Suppose that each club contains exactly g people, and each person is in exactly j clubs. Use two different ways to count the number of pairs (b, h) € P x C such that person b is in club h, and deduce a combinatorial identity.

EXERCISE 4.2.9. Let P be the set of people in a group, with |P| = p. Let C be a set of clubs formed by the people in this group, with |C| = c. Suppose that each club contains exactly g people, and each person is in exactly j clubs. Use two different ways to count the number of pairs (b, h) € P x C such that person b is in club h, and deduce a combinatorial identity.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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EXERCISE 4.2.9. Let P be the set of people in a group, with |P| = p. Let C be a set of clubs formed by the people in this group, with |C| = c. Suppose that each club contains exactly g people, and each person is in exactly j clubs. Use two different ways to count the number of pairs (b, h) € P x C such that person b is in club h, and deduce a combinatorial identity.
in Club h. and deduce a combinational identity, { (b.h) = PxC Let I = SC bin) (PxC (b is in h) Then IC PXC NI P M₂ I-SC Persen bis in clubb's bin) b (bib) h
Transcribed Image Text:in Club h. and deduce a combinational identity, { (b.h) = PxC Let I = SC bin) (PxC (b is in h) Then IC PXC NI P M₂ I-SC Persen bis in clubb's bin) b (bib) h
шишу д Use different ways of counting the number of pairs This section is read- containin {(b, h) € PxC person bis club h Solution Let in Then I = {(A,^) ( P. C | beit). we have Ic PxC J pairs (bh) such that bel K We can define maps St all pairs T₁₂: I C in 7₁: IP projection (b, h) b (forget the 27 and Coordinate (b) h "2nd projection (forget the 1st wordinate)
Transcribed Image Text:шишу д Use different ways of counting the number of pairs This section is read- containin {(b, h) € PxC person bis club h Solution Let in Then I = {(A,^) ( P. C | beit). we have Ic PxC J pairs (bh) such that bel K We can define maps St all pairs T₁₂: I C in 7₁: IP projection (b, h) b (forget the 27 and Coordinate (b) h "2nd projection (forget the 1st wordinate)
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