Examkin

Soru

Zorluk: ZorCombinations

A sports delegation of 66 athletes is to be selected from a pool of 77 sprinters and 55 distance runners. If the delegation must contain at least 44 sprinters, in how many different ways can the delegation be formed?

Cevap: 462 ways

Cevap

The total number of different ways to form the delegation is 462.
To select at least 4 sprinters in a delegation of 6 athletes, we must evaluate three mutually exclusive cases: selecting 4 sprinters and 2 distance runners (7C4×5C2=350{}^7\text{C}_4 \times {}^5\text{C}_2 = 350), selecting 5 sprinters and 1 distance runner (7C5×5C1=105{}^7\text{C}_5 \times {}^5\text{C}_1 = 105), and selecting 6 sprinters and 0 distance runners (7C6×5C0=7{}^7\text{C}_6 \times {}^5\text{C}_0 = 7). Adding these yields 350+105+7=462350 + 105 + 7 = 462 total ways.

Adım Adım Çözüm

1
Determine all valid combinations of sprinters and distance runners satisfying the condition of having at least 4 sprinters in a group of 6.
Three valid cases: (4 sprinters, 2 distance runners), (5 sprinters, 1 distance runner), and (6 sprinters, 0 distance runners).
The delegation requires 6 members and at least 4 sprinters.
2
Calculate the combinations for each case using nCr=n!r!(n−r)!{}^n\text{C}_r = \frac{n!}{r!(n-r)!}.
Case 1: 7C4×5C2=350{}^7\text{C}_4 \times {}^5\text{C}_2 = 350; Case 2: 7C5×5C1=105{}^7\text{C}_5 \times {}^5\text{C}_1 = 105; Case 3: 7C6×5C0=7{}^7\text{C}_6 \times {}^5\text{C}_0 = 7.
Apply the product rule of counting for selecting sprinters and distance runners independently within each case.
3
Sum the results of the mutually exclusive cases.
350 + 105 + 7 = 462.
Apply the addition principle of counting for mutually exclusive scenarios.

Anahtar Kavram

Combinations with restrictions and the addition principle of counting
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Combinations

Bu konudan daha fazla soru