EXERCICE 1 Sans calculatrice, calculer les nombres sui- vants: \( C_{7}^{5} ; C_{20}^{1} ; C_{5}^{4} ; A_{4}^{3} ; A_{20}^{1} ; A_{5}^{2} \). 1. A l'aide d'une calculatrice, donner: \( C_{10}^{5} ; C_{15}^{4} ; C_{13}^{6} ; A_{10}^{3} ; A_{20}^{4} ; A_{15}^{2} \). 2. Résoudre l'équation suivante dans N : \( A_{1}^{2}=6 \) 3. Développer les produits suivants: \( (x+1)^{4} ;(1-x)^{3} ;(2 x+3)^{3} \)

Here are the simplified answers for each part of the exercise: 1. **Partie 1:** - \( C_{7}^{5} = 21 \) - \( C_{20}^{1} = 20 \) - \( C_{5}^{4} = 5 \) - \( A_{4}^{3} = 24 \) - \( A_{20}^{1} = 20 \) - \( A_{5}^{2} = 20 \) 2. **Partie 2:** - \( C_{10}^{5} = 252 \) - \( C_{15}^{4} = 1365 \) - \( C_{13}^{6} = 1716 \) - \( A_{10}^{3} = 720 \) - \( A_{20}^{4} = 5,880,000 \) - \( A_{15}^{2} = 210 \) 3. **Partie 3:** - L'équation \( A_{1}^{2} = 6 \) n'a pas de solution dans \( \mathbb{N} \). 4. **Partie 4:** - \( (x+1)^{4} = x^{4} + 4x^{3} + 6x^{2} + 4x + 1 \) - \( (1-x)^{3} = -x^{3} + 3x^{2} - 3x + 1 \) - \( (2x+3)^{3} = 27 + 54x + 36x^{2} + 8x^{3} \) **Summary:** - Calculated binomial coefficients and arrangements without a calculator. - Used a calculator for larger values. - Solved the equation \( A_{1}^{2} = 6 \) and found no natural number solutions. - Expanded the given polynomial expressions.