Jasim has seven cards numbered 2 to 8 2 3 4 5 6 7 8 Jasim takes at random three of these cards. He works out the sum of the numbers on the three cards and records the result. Work out the probability that the result is an odd number. square Optional working

## Step1: Determine the total possible outcomes<br />Jasim is taking three cards out of 7. This can be written as a combination, considering that the order of the cards doesn't matter. The formula for a combination is:<br /><br />### \(C(n, r) = \frac{n!}{r!(n-r)!}\)<br /><br />Here \(n\) is the number of items you can choose from (which is 7 in this case) and \(r\) is the number of items you want to choose (which in this case is 3). <br /><br />## Step2: Calculate total possible outcomes<br />Just insert the values into the formula:<br /><br />### \(C(7, 3) = \frac{7!}{3!(7-3)!}\)<br /><br />This simplifies to:<br /><br />### \(35\)<br /><br />The total possible outcomes Jasim can have while selecting 3 cards out of 7 is 35.<br /><br />## Step3: Determine outcomes that sum to an odd number<br />To get an odd number from the sum of three numbers, there are 2 possibilities: Odd+Odd+Odd / Even+Even+Odd<br /><br />Odd numbered cards: 2 (3, 5, 7)<br />Even numbered cards: 4 (2, 4, 6, 8)<br />First possibility : Combinations of selecting all odd numbered cards <br />Odd+Odd+Odd = \(C(4, 3)\) = 4<br /><br />Second possibility : Combinations of selecting two even-numbered and one odd-numbered cards <br />Even + Even + Odd = \(C(3,2) * C(4,1) = 3 *4 = 12\)<br /><br />## Step4: Calculate probability<br />Probability = Favorable outcomes / Total possible outcomes<br /><br />Total = First possibility + Second possibility<br />Total favorable outcomes = \(4 + 12 = 16\)<br /><br />Probability = \( \frac{16}{35} \) <br /><br />The fraction simplified to it's lowest term will be \( \frac{16}{35} \)<br /><br />-------