There are 100 pigeons in a farm. Three pigeons need to be selected for random…
There are 100 pigeons in a farm. Three pigeons need to be selected for random health checkup. What are the number of possible combinations of this selection ___________
Note: Do not enter ‘,’ in the number. If number is x,yzw enter as xyzw
Answer: 161700 — The number of ways to choose r objects out of n distinct objects, when the order of selection does not matter, is given by the combination formula nCr = n! /…
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Correct answer: 161700
The number of ways to choose r objects out of n distinct objects, when the order of selection does not matter, is given by the combination formula nCr = n! / (r! × (n − r)!) — this counts unordered selections, unlike a permutation, which would also count the order of selection.
Here the total number of pigeons is n = 100 and the number to be selected is r = 3, so the required count is 100C3.
Substituting into the formula: 100C3 = 100! / (3! × 97!).
Since 100!/97! leaves only the top three factors of 100!, this reduces to (100 × 99 × 98) / (3 × 2 × 1) = (100 × 99 × 98) / 6.
Multiplying the numerator: 100 × 99 × 98 = 970200.
Dividing by 6: 970200 ÷ 6 = 161700.
Cross-check by cancelling before multiplying: 98 ÷ 2 = 49 and 99 ÷ 3 = 33, so the expression simplifies to 100 × 33 × 49 = 3300 × 49 = 161700 — the same value, confirming the result.
So the number of possible combinations of selecting 3 pigeons out of 100 is 161700.
